DERIVATIVES AND INTEGRALS

Excerpts

DERIVATIVES AND INTEGRALS
A formula such as this is called a *formula of reduction*. It is most useful when $n$ is a positive integer.
DERIVATIVES AND INTEGRALS
If $ax + by + c = 0$ then $y_{2} = 0$ (suffixes denoting differentiations with respect to $x$). We may express this by saying that *the general differential equation of all straight lines is $y_{2} = 0$*.
DERIVATIVES AND INTEGRALS
In each case we have only to write down the general equation of the curves in question, and differentiate until we have enough equations to eliminate all the arbitrary constants.
DERIVATIVES AND INTEGRALS
The constituents of a determinant are functions of $x$. Show that its differential coefficient is the sum of the determinants formed by differentiating the constituents of one row only, leaving the rest unaltered.
DERIVATIVES AND INTEGRALS
The existence of a derived function $\phi'(x)$ for all values of $x$ in the interval $a \leq x \leq b$ implies that $\phi(x)$ is continuous at every point of this interval.
DERIVATIVES AND INTEGRALS
the reader must however be careful to remember that $dy/dx$ does not mean ‘a certain number $dy$ divided by another number $dx$’: it means ‘the result of a certain operation $D_{x}$ or $d/dx$ applied to $y = \phi(x)$’, the operation being that of forming the quotient $\{\phi(x + h) - \phi(x)\}/h$ and making $h \to 0$.
DERIVATIVES AND INTEGRALS
The reader should observe that this method cannot be applied to $x^{p/q}$, where $p/q$ is a rational fraction, as we have no means of expressing $(x + h)^{p/q}$ as a finite series of powers of $h$.
DERIVATIVES AND INTEGRALS
The geometry of curves is merely one of many departments of mathematics in which the idea of a derivative finds an application.
DERIVATIVES AND INTEGRALS
The notion of ‘velocity’ is in fact merely a special case of that of the derivative of a function.
DERIVATIVES AND INTEGRALS
If $\phi(a) = 0$ and $\phi(b) = 0$, then there must be at least one value of $x$ which lies between $a$ and $b$ and for which $\phi'(x) = 0$.
DERIVATIVES AND INTEGRALS
This function is called the *second derivative* or *second differential coefficient* of $\phi(x)$.
DERIVATIVES AND INTEGRALS
A **** condition for a maximum or minimum value of $\phi(x)$ at $x = \xi$ is that $\phi'(\xi) = 0$.
DERIVATIVES AND INTEGRALS
There is no derivative for $x = 0$, and no tangent to the graph at $P$.
DERIVATIVES AND INTEGRALS
The reader, if he considers what the question means and tries to answer it in the light of common sense, will probably incline to the answer *No*. It is, however, not difficult to see that this answer is wrong.
DERIVATIVES AND INTEGRALS
But there is no practical difficulty in the actual calculation of the derivative of such a function: the method to be adopted will be illustrated sufficiently by an example.
DERIVATIVES AND INTEGRALS
It will then follow by the principle of mathematical induction that $a_{n, r} = \dbinom{n}{r}$ for all values of $n$ and $r$ in question.
DERIVATIVES AND INTEGRALS
Of course from a geometrical point of view the result is intuitive, the inequality $\phi'(x) > 0$ expressing the fact that the tangent to the curve $y = \phi(x)$ makes a positive acute angle with the axis of $x$.
DERIVATIVES AND INTEGRALS
It is natural to consider the converse question, that of *determining a function whose derivative is a given function*.
DERIVATIVES AND INTEGRALS
There is however one case of exception to the first formula, that in which $m = -1$.
DERIVATIVES AND INTEGRALS
the $\int$ and the $dx$ no more mean anything when taken by themselves than do the $d$ and $dx$ of the other operative symbol $d/dx$.
DERIVATIVES AND INTEGRALS
If the equation $Q(x) = 0$ cannot be solved algebraically, then the method of partial fractions naturally fails and recourse must be had to other methods.
DERIVATIVES AND INTEGRALS
Suppose, for example, that $\phi(x) = x\psi(x)$, where $\psi(x)$ is the second derivative of a known function $\chi(x)$.
DERIVATIVES AND INTEGRALS
It is indeed one which needs and has received the most careful mathematical analysis: later on we shall return to it and explain precisely what is meant by ascribing an ‘area’ to such a region of space as $ONPP_{0}$.
DERIVATIVES AND INTEGRALS
Calculate $\Phi(x)$, the integral of $\phi(x)$. This involves an arbitrary constant, which we suppose so chosen that $\Phi(0) = 0$.
DERIVATIVES AND INTEGRALS
It is however easy to see what the *formula* must be.
DERIVATIVES AND INTEGRALS
This integral cannot however be evaluated in terms of such functions as are at present at our disposal.
DERIVATIVES AND INTEGRALS
Put $x + \frac{1}{2}p = t$, $q - \frac{1}{4}p^{2} = \lambda$: then we obtain
DERIVATIVES AND INTEGRALS
This equation has three real roots if $s^{4} > 27\Delta^{2}$, and one in the contrary case.
DERIVATIVES AND INTEGRALS
If $\phi(x) \to a$ as $x \to \infty$, then $\phi'(x)$ cannot tend to any limit other than zero.
DERIVATIVES AND INTEGRALS
This theorem reduces to the Mean Value Theorem ([§]125) when $\phi(x) = x$ and $\psi(x) = 1$.
DERIVATIVES AND INTEGRALS
In an equilateral triangle (the triangle of minimum perimeter for a given area) $s^{4} = 27\Delta^{2}$; thus it is impossible that $s^{4} < 27\Delta^{2}$.
DERIVATIVES AND INTEGRALS
The reader will probably remember that in elementary geometry the tangent to a curve at $P$ is defined to be ‘the limiting position of the chord $PQ$, when $Q$ moves up towards coincidence with $P$’.
DERIVATIVES AND INTEGRALS
But $\phi(x)$ has no derivative for $x = 0$. For $\phi'(0)$ would be, by definition, $\lim\{\phi(h) - \phi(0)\}/h$ or $\lim\sin(1/h)$; and no such limit exists.
DERIVATIVES AND INTEGRALS
The notion of a derivative or differential coefficient was suggested to us by geometrical considerations. But there is nothing geometrical in the notion itself.
DERIVATIVES AND INTEGRALS
Of these the last is the most usual and convenient: the reader must however be careful to remember that $dy/dx$ does not mean ‘a certain number $dy$ divided by another number $dx$’: it means ‘the result of a certain operation $D_{x}$ or $d/dx$ applied to $y = \phi(x)$’, the operation being that of forming the quotient $\{\phi(x + h) - \phi(x)\}/h$ and making $h \to 0$.
DERIVATIVES AND INTEGRALS
These differences may be called the *increments* of $x$ and $y$ respectively, and denoted by $\delta x$ and $\delta y$.
DERIVATIVES AND INTEGRALS
Incidentally we have proved that *the derivative of $x^{m}$ is $mx^{m-1}$, for all integral values of $m$ positive or negative*.
DERIVATIVES AND INTEGRALS
We have seen already ([§]117) that the derivative of this function is $mx^{m-1}$ when $m$ is an integer positive or negative; and we shall now prove that this result is true for all rational values of $m$.
DERIVATIVES AND INTEGRALS
The differentiation of *implicit* algebraical functions involves certain theoretical difficulties to which we shall return in Ch.VII. But there is no practical difficulty in the actual calculation of the derivative of such a function: the method to be adopted will be illustrated sufficiently by an example.
DERIVATIVES AND INTEGRALS
An immediate deduction from Theorem A is the following important theorem, generally known as Rolle’s Theorem. In view of the great importance of this theorem it may be well to repeat that its truth depends on the assumption of the existence of the derivative $\phi'(x)$ for all values of $x$ in question.
DERIVATIVES AND INTEGRALS
Thus if $y = x^{3}$ then $\phi'(x) = 3x^{2}$, which vanishes when $x = 0$. But $x = 0$ does not give either a maximum or a minimum of $x^{3}$, as is obvious from the form of the graph of $x^{3}$ ([fig:10]Fig. 10, p.45).
DERIVATIVES AND INTEGRALS
If the sign of $\phi'(x)$ changes at $x = \xi$ from positive to negative, then $x = \xi$ gives a maximum of $\phi(x)$: and if the sign of $\phi'(x)$ changes in the opposite sense, then $x = \xi$ gives a minimum.
DERIVATIVES AND INTEGRALS
Suppose, *e.g.*, that $\phi''(\xi) < 0$. Then, by Theorem A, $\phi'(x)$ is negative when $x$ is less than $\xi$ but sufficiently near to $\xi$, and positive when $x$ is greater than $\xi$ but sufficiently near to $\xi$. Thus $x = \xi$ gives a maximum.
DERIVATIVES AND INTEGRALS
*Can* a function $\phi(x)$ have a derivative for all values of $x$ which is not itself continuous? In other words can a curve have a tangent at every point, and yet the direction of the tangent not vary continuously? The reader, if he considers what the question means and tries to answer it in the light of common sense, will probably incline to the answer *No*. It is, however, not difficult to see that this answer is wrong.
DERIVATIVES AND INTEGRALS
The theorem of *integration by parts* is merely another way of stating the rule for the differentiation of a product proved in [§]113.
DERIVATIVES AND INTEGRALS
The integral of any rational function of $\cos x$ and $\sin x$ may be calculated by the substitution $\tan \frac{1}{2}x = t$.
DERIVATIVES AND INTEGRALS
It is easy to see that if we can find the integral of $y = f(x)$ then we can always find that of $x = \phi(y)$, where $\phi$ is the function inverse to $f$.
DERIVATIVES AND INTEGRALS
Thus *the ordinate of the curve is the derivative of the area, and the area is the integral of the ordinate*.
DERIVATIVES AND INTEGRALS
The reader is of course familiar with the idea of an ‘area’, and in particular with that of an area such as $ONPP_{0}$. This idea we shall at present take for granted.
DERIVATIVES AND INTEGRALS
The notion of the length of a curve, other than a straight line, is in reality a more difficult one even than that of an area.
DERIVATIVES AND INTEGRALS
The explanation of this is of course that between $x = \pi$ and $x = 2\pi$ the curve lies below the axis of $x$, and so the corresponding part of the area is counted negative in applying the method.
DERIVATIVES AND INTEGRALS
The integrals of the inverse sine and tangent and of the logarithm can easily be calculated by integration by parts.
DERIVATIVES AND INTEGRALS
Before we give a strict proof of this theorem, which is perhaps the most important theorem in the Differential Calculus, it will be well to point out its obvious geometrical meaning.
DERIVATIVES AND INTEGRALS
For $\phi'(\xi)$ is the tangent of the angle which the tangent at $P$ makes with $OX$, and $\{\phi(b) - \phi(a)\}/(b - a)$ the tangent of the angle which $AB$ makes with $OX$.
DERIVATIVES AND INTEGRALS
It should be observed that it has not been assumed in this proof that $\phi'(x)$ is continuous.
DERIVATIVES AND INTEGRALS
In the first place we want to know whether such a function as $\phi(x)$ *actually exists*. This question must be carefully distinguished from the question as to whether (supposing that there is such a function) we can find any simple formula to express it.
DERIVATIVES AND INTEGRALS
Whether there are continuous functions which *never* have derivatives, or continuous curves which never have tangents, is a further question which is at present beyond us. Common-sense says *No*: but, as we have already stated in [§]111, this is one of the cases in which higher mathematics has proved common-sense to be mistaken.
DERIVATIVES AND INTEGRALS
It is hardly necessary to point out that $\int\dots dx$ like $d/dx$ must, at present at any rate, be regarded purely as a symbol of operation: the $\int$ and the $dx$ no more mean anything when taken by themselves than do the $d$ and $dx$ of the other operative symbol $d/dx$.
DERIVATIVES AND INTEGRALS
These formulae must be understood as meaning that the function on the right-hand side is *one* integral of that under the sign of integration. The *most general* integral is of course obtained by adding to the former a constant $C$, known as the **constant** of integration.
DERIVATIVES AND INTEGRALS
Thus the integral of $R(\sqrt{x})$, where $R$ denotes a rational function, is reduced by the substitution $x = t^{2}$ to the integral of $2tR(t^{2})$, *i.e.* to the integral of a rational function of $t$. This method of integration is called **by rationalisation**, and is of extremely wide application.

Equations

DERIVATIVES AND INTEGRALS
\lim_{h \to 0} \frac{\phi(x + h) - \phi(x)}{h} = \tan\psi
The limit of the difference quotient of phi at x, as h tends to zero, equals the tangent of the angle psi that the tangent at P makes with OX.
DERIVATIVES AND INTEGRALS
\phi'(x) = \lim_{h \to 0} \frac{\phi(x + h) - \phi(x)}{h}
The derivative of phi at x is defined as the limit of the difference quotient as h tends to zero, without any geometrical picture.
DERIVATIVES AND INTEGRALS
\phi(x) = a_{0}x^{n} + a_{1}x^{n-1} + \dots + a_{n}
A polynomial of degree n in x written as a sum of powers of x with constant coefficients.
DERIVATIVES AND INTEGRALS
\phi'(x) = na_{0}x^{n-1} + (n - 1)a_{1}x^{n-2} + \dots + a_{n-1}
The derivative of a polynomial of degree n is obtained by multiplying each term's coefficient by its power and lowering that power by one.
DERIVATIVES AND INTEGRALS
\phi'(x) = n \left\{ a_{0}x^{n-1} + \binom{n - 1}{1} a_{1}x^{n-2} + \binom{n - 1}{2} a_{2}x^{n-3} + \dots + a_{n-1} \right\}
The derivative of a polynomial written in binomial form, with binomial coefficients, is n times a polynomial of degree n-1 of the same binomial type.
DERIVATIVES AND INTEGRALS
\phi(x) = a_{0}(x - \alpha_{1})(x - \alpha_{2}) \dots (x - \alpha_{n})
A polynomial of degree n factorises into n linear factors, with real or complex roots alpha.
DERIVATIVES AND INTEGRALS
\phi'(x) = a_{0}\tsum (x - \alpha_{2})(x - \alpha_{3}) \dots (x - \alpha_{n})
The derivative of a factored polynomial is a_0 times the sum of all products of n-1 of its linear factors.
DERIVATIVES AND INTEGRALS
\phi'(x) = a_{0} \tsum m_{1}(x - \alpha_{1})^{m_{1}-1} (x - \alpha_{2})^{m_{2}}\dots (x - \alpha_{\nu})^{m_{\nu}}
The derivative of a polynomial with repeated roots, each factor raised to multiplicity m, is a sum over the roots with the multiplicity lowering by one for the differentiated factor.
This equation is in FUNCTIONS OF REAL VARIABLES (FUNCTIONS OF REAL VARIABLES)
DERIVATIVES AND INTEGRALS
R'(x) = \frac{P'(x)Q(x) - P(x)Q'(x)}{\{Q(x)\}^{2}}
The derivative of a quotient of two polynomials equals (P'Q - PQ') divided by Q squared.
DERIVATIVES AND INTEGRALS
-\frac{pA(x -\alpha)^{p-1}}{(x - \alpha)^{2p}} = -\frac{pA}{(x - \alpha)^{p+1}}
The derivative of the partial-fraction term A over (x - alpha) to the power p is -pA over (x - alpha) to the power p+1.
DERIVATIVES AND INTEGRALS
\phi'(x) = f'(x) + F'(x)
The derivative of a sum of two differentiable functions is the sum of their derivatives.
DERIVATIVES AND INTEGRALS
\phi'(x) = kf'(x)
The derivative of a constant multiple of a function is the constant times the derivative of the function.
DERIVATIVES AND INTEGRALS
\phi'(x) = f(x)F'(x) + f'(x)F(x)
The derivative of a product of two differentiable functions is f times F-prime plus f-prime times F.
DERIVATIVES AND INTEGRALS
\phi'(x) = -\frac{f'(x)}{\{f(x)\}^{2}}
The derivative of the reciprocal of a function is minus its derivative divided by the square of the function, where f is non-zero.
DERIVATIVES AND INTEGRALS
\phi'(x) = \frac{f'(x)F(x) - f(x)F'(x)}{\{F(x)\}^{2}}
The derivative of a quotient f over F is (f'F - fF') divided by F squared.
DERIVATIVES AND INTEGRALS
\phi'(x) = F'\{f(x)\} f'(x)
The derivative of a composite function F of f(x) is F-prime evaluated at f(x), times f-prime(x) (chain rule).
DERIVATIVES AND INTEGRALS
\phi'(x) = \frac{1}{\psi'(y)}
The derivative of the inverse function phi is the reciprocal of the derivative of psi, evaluated at y = phi(x).
DERIVATIVES AND INTEGRALS
\frac{dy}{dx} = \frac{dy_{1}}{dx} + \frac{dy_{2}}{dx}
In differential notation, the derivative of a sum y = y1 + y2 is the sum of the derivatives.
DERIVATIVES AND INTEGRALS
\frac{dy}{dx} = k\frac{dy_{1}}{dx}
In differential notation, the derivative of k times y1 is k times the derivative of y1.
DERIVATIVES AND INTEGRALS
\frac{dy}{dx} = y_{1}\frac{dy_{2}}{dx} + y_{2}\frac{dy_{1}}{dx}
In differential notation, the derivative of a product y = y1 y2 is y1 times dy2/dx plus y2 times dy1/dx.
DERIVATIVES AND INTEGRALS
\frac{dy}{dx} = -\frac{1}{y_{1}^{2}}\, \frac{dy_{1}}{dx}
In differential notation, the derivative of 1/y1 is minus dy1/dx divided by y1 squared.
DERIVATIVES AND INTEGRALS
\frac{dy}{dx} = \biggl(y_{2}\frac{dy_{1}}{dx} - y_{1}\frac{dy_{2}}{dx}\biggr) \bigg/ y_{2}^{2}
In differential notation, the derivative of the quotient y1/y2 is (y2 dy1/dx - y1 dy2/dx) divided by y2 squared.
DERIVATIVES AND INTEGRALS
\frac{dz}{dx} = \frac{dz}{dy}\, \frac{dy}{dx}
In differential notation, the chain rule: dz/dx equals dz/dy times dy/dx when z is a function of y and y of x.
DERIVATIVES AND INTEGRALS
\dfrac{dy}{dx} = 1 \bigg/ \biggl(\dfrac{dx}{dy}\biggr)
The derivative of y with respect to x is the reciprocal of the derivative of x with respect to y, for an inverse function.
DERIVATIVES AND INTEGRALS
y - y_{0} = (x - x_{0}) \phi'(x_{0})
The tangent to the curve y = phi(x) at (x0, y0) is the line through that point with slope phi'(x0).
DERIVATIVES AND INTEGRALS
(y - y_{0}) \phi'(x_{0}) + x - x_{0} = 0
The normal at (x0, y0) is the line through that point perpendicular to the tangent.
DERIVATIVES AND INTEGRALS
\frac{dy}{dx} = \biggl(\frac{dy}{dz}\biggr) \bigg/ \biggl(\frac{dx}{dz}\biggr) = \frac{p}{q} z^{p-q} = mx^{m-1}
For y = x^m with m = p/q, the derivative with respect to x is mx^(m-1), obtained through the substitution z = x^(1/q) and the chain rule.
DERIVATIVES AND INTEGRALS
\phi'(x) = \lim_{h \to 0} \frac{(x + h)^{m} - x^{m}}{h}
The derivative of phi(x) is the limit of the difference quotient as h tends to zero.
DERIVATIVES AND INTEGRALS
\lim_{\xi \to x} \frac{\xi^{m} - x^{m}}{\xi - x} = mx^{m-1}
The limit of the difference quotient taken with xi tending to x gives the derivative mx^(m-1) of x^m.
DERIVATIVES AND INTEGRALS
\frac{d}{dx} (ax + b)^{m} = ma(ax + b)^{m-1}
The derivative of (ax + b) raised to the power m is ma(ax + b)^(m-1), valid for all rational m.
DERIVATIVES AND INTEGRALS
x^{3} + y^{3} - 3axy = 0
The implicit relation between x and y used as the example for differentiating an implicit algebraic function.
DERIVATIVES AND INTEGRALS
x^{2} + y^{2} \frac{dy}{dx} - a\left(y + x \frac{dy}{dx}\right) = 0
The result of differentiating x^3 + y^3 - 3axy = 0 with respect to x, which contains dy/dx.
DERIVATIVES AND INTEGRALS
\frac{dy}{dx} = -\frac{x^{2} - ay}{y^{2} - ax}
The derivative of the implicitly defined y with respect to x, found by solving the differentiated equation.
DERIVATIVES AND INTEGRALS
D_{x} \sin x = \cos x
The derivative of sin x with respect to x is cos x.
DERIVATIVES AND INTEGRALS
D_{x} \cos x = -\sin x
The derivative of cos x with respect to x is minus sin x.
DERIVATIVES AND INTEGRALS
D_{x} \tan x = \sec^{2} x
The derivative of tan x is sec^2 x.
DERIVATIVES AND INTEGRALS
D_{x} \cot x = -\cosec^{2} x
The derivative of cot x is minus cosec^2 x.
DERIVATIVES AND INTEGRALS
D_{x} \sec x = \tan x \sec x
The derivative of sec x is tan x sec x.
DERIVATIVES AND INTEGRALS
D_{x} \cosec x = -\cot x\cosec x
The derivative of cosec x is minus cot x cosec x.
DERIVATIVES AND INTEGRALS
D_{x} \arcsin x = ±1/\sqrtp{1 - x^{2}}
The derivative of the inverse sine is 1 over the square root of 1 - x^2, with the sign fixed by cos(arcsin x).
DERIVATIVES AND INTEGRALS
D_{x} \arccos x = \mp 1/\sqrtp{1 - x^{2}}
The derivative of the inverse cosine is minus 1 over the square root of 1 - x^2, with the sign fixed by sin(arccos x).
DERIVATIVES AND INTEGRALS
D_{x} \arctan x = 1/(1 + x^{2})
The derivative of the inverse tangent is 1 over (1 + x^2).
DERIVATIVES AND INTEGRALS
D_{x} \arccot x = -1/(1 + x^{2})
The derivative of the inverse cotangent is minus 1 over (1 + x^2).
DERIVATIVES AND INTEGRALS
D_{x} \arcsec x = ± 1/\{x\sqrtp{x^{2} - 1}\}
The derivative of the inverse secant is ±1 over x times the square root of x^2 - 1.
DERIVATIVES AND INTEGRALS
D_{x} \arccosec x = \mp 1/\{x\sqrtp{x^{2} - 1}\}
The derivative of the inverse cosecant is minus or plus 1 over x times the square root of x^2 - 1, with a sign convention.
DERIVATIVES AND INTEGRALS
D_{x} \arcsin(x/a) = ±1/\sqrtp{a^{2} - x^{2}}
The more general derivative of the inverse sine of x/a, with the sign given by a cos{arcsin(x/a)}.
DERIVATIVES AND INTEGRALS
D_{x} \arctan(x/a) = a/(x^{2} + a^{2})
The derivative of the inverse tangent of x/a is a over (x^2 + a^2).
DERIVATIVES AND INTEGRALS
a\sqrtb{1 - (x^{2}/a^{2})} = ±\sqrtp{a^{2} - x^{2}}
The square root of 1 - x^2/a^2 times a equals plus or minus the square root of a^2 - x^2, according as a is positive or negative.
DERIVATIVES AND INTEGRALS
\phi'(x) = 0
Rolle's theorem: if phi vanishes at a and b, then phi' vanishes at some point between a and b.
DERIVATIVES AND INTEGRALS
\phi'(x) > 0
If the derivative is positive throughout an interval, phi is an increasing function throughout that interval, in the stricter sense.
DERIVATIVES AND INTEGRALS
\phi'(\xi) = 0
A necessary condition for a maximum or minimum of phi at x = xi is that the derivative vanishes at xi.
DERIVATIVES AND INTEGRALS
\phi'(x) = 3x^{2}
The derivative of y = x^3 is 3x^2, which vanishes at x = 0 without giving a maximum or minimum there.
DERIVATIVES AND INTEGRALS
y = 1 - \sqrtp{x^{2}}
The example function, with the positive square root, for which Rolle's theorem fails because there is no derivative at x = 0.
DERIVATIVES AND INTEGRALS
\phi(x) = x^{2}\sin(1/x)
Definition of a function that has a derivative everywhere but whose derivative is discontinuous at x = 0.
DERIVATIVES AND INTEGRALS
\phi'(x) = 2x \sin(1/x) - \cos(1/x)
The derivative of x^2 sin(1/x) for x not equal to zero, which oscillates near zero and so is discontinuous at x = 0.
DERIVATIVES AND INTEGRALS
\phi'(0) = \lim_{h \to 0} \frac{h^{2}\sin(1/h)}{h} = 0
The derivative at zero exists and equals zero, obtained as a limit of the difference quotient.
DERIVATIVES AND INTEGRALS
\phi(x) = x^{2}\sin(1/x) + ax
The modified example function with a linear term, which has positive derivative at zero but is not steadily increasing on any interval containing zero.
DERIVATIVES AND INTEGRALS
\phi'(x) = 2x\sin(1/x) - \cos(1/x) + a
The derivative of the modified example for x not equal to zero, which oscillates between a - 1 and a + 1 as x tends to zero.
DERIVATIVES AND INTEGRALS
\log (1/x) = -\log x
The logarithm of the reciprocal of x is minus the logarithm of x.
DERIVATIVES AND INTEGRALS
\phi(b) - \phi(a) = (b - a)\phi'(\xi)
If φ has a derivative throughout the interval from a to b, then at some value ξ between a and b the derivative equals the average rate of change over the interval.
DERIVATIVES AND INTEGRALS
\phi(b) = \phi(a) + (b - a) \phi'\{a + \theta(b - a)\}
The mean value theorem restated: φ(b) equals φ(a) plus (b − a) times the derivative at some point a + θ(b − a), with θ between 0 and 1.
DERIVATIVES AND INTEGRALS
\phi(a + h) = \phi(a) + h\phi'(a + \theta h)
The mean value theorem in increment form: φ(a+h) equals φ(a) plus h times the derivative at a point between a and a+h.
DERIVATIVES AND INTEGRALS
\phi(x) = \int \psi(x)\, dx
φ is an integral (integral function) of ψ, meaning φ'(x) = ψ(x); this is the notation for integration.
DERIVATIVES AND INTEGRALS
\int x^{m}\, dx = \frac{x^{m+1}}{m + 1}
The integral of x to the power m is x to the power m+1 divided by m+1, for m not equal to −1.
DERIVATIVES AND INTEGRALS
\int \cos x\, dx = \sin x
The integral of cos x is sin x (one integral; the arbitrary constant C may be added).
DERIVATIVES AND INTEGRALS
\int \sin x\, dx = -\cos x
The integral of sin x is minus cos x (one integral; the arbitrary constant C may be added).
DERIVATIVES AND INTEGRALS
\int \frac{dx}{x} = \log x
For positive x, the integral of 1/x is the logarithmic function log x, which is defined by this equation.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{x} = \log(-x)
For negative x, the integral of 1/x is log(−x), since the derivative of log(−x) is 1/x.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{x} = \log(±x) = \log|x|
Combining the positive and negative cases, the integral of 1/x is log|x| for all real x other than zero, where the ambiguous sign is chosen to make ±x positive.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{x} = \tfrac{1}{2}\log x^{2}
Equivalent single form for the integral of 1/x, since log x² equals 2 log|x|.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{1 + x^{2}} = \arctan x
The integral of 1/(1 + x²) is the inverse tangent of x.
DERIVATIVES AND INTEGRALS
\int \frac{x}{\sqrtp{1 - x^{2}}} = ±\arcsin x
The integral of x over the square root of 1 − x² is ± arcsin x, the sign being fixed by the rule given in section 119.
DERIVATIVES AND INTEGRALS
\log 1 = 0
The logarithm of 1 is zero.
DERIVATIVES AND INTEGRALS
\log xy = \log x + \log y
The logarithm of a product is the sum of the logarithms of the factors.
DERIVATIVES AND INTEGRALS
\int \{f(x) + F(x)\}\, dx = \int f(x) dx + \int F(x)\, dx
The integral of a sum is the sum of the integrals, up to the arbitrary constants.
DERIVATIVES AND INTEGRALS
\int kf(x)\, dx = k\int f(x)\, dx
A constant factor can be taken outside the integral sign.
DERIVATIVES AND INTEGRALS
\int (a_{0}x^{n} + a_{1}x^{n-1} + \dots + a_{n})\, dx = \frac{a_{0}x^{n+1}}{n + 1} + \frac{a_{1}x^{n}}{n} + \dots + a_{n}x
The integral of a polynomial is obtained term by term, each power x^k becoming x^(k+1)/(k+1).
DERIVATIVES AND INTEGRALS
\int \frac{A}{(x - \alpha)^{p}}\, dx = -\frac{A}{p - 1}\, \frac{1}{(x - \alpha)^{p-1}}
The integral of A over (x − α) to the power p, for p not equal to 1, is minus A/(p−1) times 1/(x−α) to the power p−1; this holds whether α is real or complex.
DERIVATIVES AND INTEGRALS
\int F'\{f(x)\}\, f'(x)\, dx = F\{f(x)\}
The integral of the derivative of F at f(x), multiplied by the derivative of f, is F of f(x): the chain rule read backwards.
DERIVATIVES AND INTEGRALS
\int \psi(ax + b)\, dx = \frac{1}{a}\phi(ax + b)
If φ is an integral of ψ, then the integral of ψ(ax+b) is φ(ax+b) divided by a.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{ax + b} = \frac{1}{a} \log|ax + b|
The integral of 1/(ax+b) is log of the absolute value of ax+b, divided by a.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{x - \alpha} = \log|x - \alpha|
For real α, the integral of 1/(x−α) is the logarithm of the absolute value of x−α.
DERIVATIVES AND INTEGRALS
\lambda = A/2a
The constant λ in the partial fraction form equals A divided by 2a.
DERIVATIVES AND INTEGRALS
\mu = -D/(2a\sqrt{\Delta})
The constant μ in the combined partial fraction equals minus D over 2a times the square root of Δ.
DERIVATIVES AND INTEGRALS
\gamma = -b/a
The real part γ of the complex root equals minus b over a.
DERIVATIVES AND INTEGRALS
\delta = \sqrt{\Delta}/a
The imaginary part δ of the complex root equals the square root of Δ over a.
DERIVATIVES AND INTEGRALS
\Delta = ac - b^{2}
Δ is defined as ac minus b squared for the quadratic ax² + 2bx + c.
DERIVATIVES AND INTEGRALS
D = aB - bA
D is defined as aB minus bA, a combination of the coefficients of the numerator Ax + B and the quadratic.
DERIVATIVES AND INTEGRALS
\int \frac{f'(x)}{f(x)}\, dx = \log |f(x)|
The integral of a logarithmic derivative f'(x)/f(x) is the logarithm of the absolute value of f(x).
DERIVATIVES AND INTEGRALS
\int \frac{2(x - \lambda)}{(x - \lambda)^{2} + \mu^{2}}\, dx = \log\{(x - \lambda)^{2} + \mu^{2}\}
The integral of 2(x−λ)/((x−λ)²+μ²) is the logarithm of (x−λ)²+μ².
DERIVATIVES AND INTEGRALS
\int \frac{-2\delta\mu}{(x - \lambda)^{2} + \mu^{2}}\, dx = -2\delta \arctan \left(\frac{x - \lambda}{\mu}\right)
The integral of −2δμ/((x−λ)²+μ²) is −2δ times the inverse tangent of (x−λ)/μ.
DERIVATIVES AND INTEGRALS
ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0
The general equation of the second degree in x and y, whose graph is a conic section.
DERIVATIVES AND INTEGRALS
aX^{2} + 2hXY + bY^{2} + 2GX + 2FY = 0
The conic written in the shifted variables X = x − ξ and Y = y − η.
DERIVATIVES AND INTEGRALS
F = h\xi + b\eta + f
The coefficient F is defined as hξ + bη + f for a point (ξ, η) on the conic.
DERIVATIVES AND INTEGRALS
G = a\xi + h\eta + g
The coefficient G is defined as aξ + hη + g for a point (ξ, η) on the conic.
DERIVATIVES AND INTEGRALS
x - \xi = -\frac{2 (G + Ft)}{a + 2ht + bt^{2}}
Rational parametrisation of x in terms of the parameter t = Y/X along the conic.
DERIVATIVES AND INTEGRALS
y - \eta = -\frac{2t(G + Ft)}{a + 2ht + bt^{2}}
Rational parametrisation of y in terms of the parameter t along the conic.
DERIVATIVES AND INTEGRALS
hx + by + f = -\tfrac{1}{2}(a + 2ht + bt^{2}) \frac{dx}{dt}
The linear expression hx + by + f equals minus half of the quadratic in t times dx/dt.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{hx + by + f}= -2\int \frac{dt}{a + 2ht + bt^{2}}
The integral over the conic of dx/(hx+by+f) equals minus two times the integral of dt over the quadratic in t.
DERIVATIVES AND INTEGRALS
y^{2} = ax^{2} + 2bx + c
The curve y² = ax² + 2bx + c, the graph of y as a function of x, is treated as a conic in the integrals of section 135.
DERIVATIVES AND INTEGRALS
2\frac{dx}{dt} = \frac{(t^{2} + c)\sqrt{a} + 2bt}{(t\sqrt{a} + b)^{2}}
Derivative of x with respect to t after the substitution y + x√a = t.
DERIVATIVES AND INTEGRALS
2y = \frac{(t^{2} + c)\sqrt{a} + 2bt}{t\sqrt{a} + b}
y expressed as a rational function of t after the substitution y + x√a = t.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{y} = \int \frac{dt}{t\sqrt{a} + b} = \frac{1}{\sqrt{a}} \log \left|x\sqrt{a} + y + \frac{b}{\sqrt{a}}\right|
The integral of dx/y, for y² = ax² + 2bx + c with a > 0, equals (1/√a) log of the absolute value of x√a + y + b/√a.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{\sqrtp{x^{2} + a^{2}}} = \log \{x + \sqrtp{x^{2} + a^{2}}\}
The integral of 1/√(x²+a²) is the logarithm of x plus √(x²+a²).
DERIVATIVES AND INTEGRALS
\int \frac{dx}{\sqrtp{x^{2} - a^{2}}} = \log |x + \sqrtp{x^{2} - a^{2}}|
The integral of 1/√(x²−a²) is the logarithm of the absolute value of x + √(x²−a²).
DERIVATIVES AND INTEGRALS
\int \frac{dx}{\sqrtp{a^{2} - x^{2}}} = \arcsin(x/a)
The integral of 1/√(a²−x²) is the inverse sine of x/a.
DERIVATIVES AND INTEGRALS
\lambda x + \mu = (\lambda/a) (ax + b) + \mu - (\lambda b/a)
Rewrites the linear numerator λx + μ in terms of ax + b, to reduce the integral in section 136.
DERIVATIVES AND INTEGRALS
\int \frac{ax + b}{\sqrtp{ax^{2} + 2bx + c}}\, dx = \sqrtp{ax^{2} + 2bx + c}
The integral of (ax+b)/√(ax²+2bx+c) is √(ax²+2bx+c).
DERIVATIVES AND INTEGRALS
\int \frac{(\lambda x + \mu)\, dx}{\sqrtp{ax^{2} + 2bx + c}} = \frac{\lambda}{a} \sqrtp{ax^{2} + 2bx + c} + \left(\mu - \frac{\lambda b}{a}\right) \int \frac{dx}{\sqrtp{ax^{2} + 2bx + c}}
The integral of (λx+μ)/√(ax²+2bx+c) reduces to a square-root term plus a constant multiple of the integral of 1/√(ax²+2bx+c).
DERIVATIVES AND INTEGRALS
\kappa = (ac - b^{2})/a
κ is defined as (ac − b²)/a, the constant that appears after the substitution x√a + b/√a = t.
DERIVATIVES AND INTEGRALS
\int(\lambda x + \mu) \sqrtp{ax^{2} + 2bx + c}\, dx \\ = \left(\frac{\lambda}{3a}\right) (ax^{2} + 2bx + c)^{3/2} + \left(\Add{\mu} - \frac{\lambda b}{a}\right) \int \sqrtp{ax^{2} + 2bx + c}\, dx
The integral of (λx+μ)√(ax²+2bx+c) equals a (3/2)-power term plus a constant multiple of the integral of √(ax²+2bx+c).
DERIVATIVES AND INTEGRALS
\int R(x, \sqrt{X})\, dx
The most general integral of a real rational function R of x and the square root of X, where X = y^2 = ax^2 + 2bx + c; this is equation (1).
DERIVATIVES AND INTEGRALS
\int f'(x)F(x)\, dx = f(x)F(x) - \int f(x)F'(x)\, dx
The integral of f'(x) times F(x) equals f(x)F(x) minus the integral of f(x) times F'(x).
DERIVATIVES AND INTEGRALS
\int\phi(x)\, dx = \int x\chi''(x)\, dx = x\chi'(x) - \int \chi'(x)\, dx = x\chi'(x) - \chi(x)
Worked case of integration by parts: when phi(x) = x times the second derivative of chi, the integral of phi is x chi'(x) minus chi(x).
DERIVATIVES AND INTEGRALS
F(x) = \sqrtp{ax^{2} + 2bx + c} = y
In the worked illustration F(x) is the square root of the quadratic in x, and this is written as y.
DERIVATIVES AND INTEGRALS
\int y\, dx = \frac{(ax + b)y}{2a} + \frac{ac - b^{2}}{2a} \int \frac{dx}{y}
Integrating by parts reduces the integral of y dx, where y is the square root of a quadratic, to the integral of 1/y.
DERIVATIVES AND INTEGRALS
\frac{A + B\sqrt{X}}{C + D\sqrt{X}} = \frac{(A + B\sqrt{X})(C - D\sqrt{X})}{C^{2} - D^{2}X} = E + F\sqrt{X}
Multiplying numerator and denominator by C minus D sqrt(X) reduces a quotient of this form to E plus F sqrt(X), with E and F rational in x.
DERIVATIVES AND INTEGRALS
\int \frac{G}{\sqrt{X}}\, dx
The one remaining type of integral, which can always be evaluated by splitting G into partial fractions; this is equation (2).
DERIVATIVES AND INTEGRALS
\int \frac{x^{m}}{\sqrt{X}}\, dx
Type (i) integral (3): x to the power m, with m a positive integer, divided by sqrt(X).
DERIVATIVES AND INTEGRALS
\frac{d}{dx}(x^{m-1}\sqrt{X}) = (m - 1)x^{m-2} \sqrt{X} + \frac{(ax + b) x^{m-1}}{\sqrt{X}} = \frac{\alpha x^{m} + \beta x^{m-1} + \gamma x^{m-2}}{\sqrt{X}}
Differentiating x^(m-1) sqrt(X) gives a combination of three successive terms over sqrt(X), so integrating yields a relation between three successive integrals of type (3).
DERIVATIVES AND INTEGRALS
\int \frac{dx}{(x - p)^{m}\sqrt{X}}
Type (ii) integral (4), where p is real; the substitution x - p = 1/t reduces it to a type (3) integral in t.
DERIVATIVES AND INTEGRALS
\int \frac{Lx + M}{(Ax^{2} + 2Bx + C) \sqrt{ax^{2} + 2bx + c}}\, dx
Type (iii) integral (5), arising from a pair of conjugate complex roots of the denominator of G.
DERIVATIVES AND INTEGRALS
x = \frac{\mu t + \nu}{t + 1}
The substitution used to evaluate integral (5), with mu and nu chosen to satisfy the two conditions given.
DERIVATIVES AND INTEGRALS
a\mu\nu + b(\mu + \nu) + c = 0
First condition that fixes the constants mu and nu in the substitution for integral (5).
DERIVATIVES AND INTEGRALS
A\mu\nu + B(\mu + \nu) + C = 0
Second condition fixing mu and nu, using the coefficients of the quadratic factor of the denominator.
DERIVATIVES AND INTEGRALS
(aB - bA)\xi^{2} - (cA - aC)\xi + (bC - cB) = 0
Quadratic equation whose roots are mu and nu; the book notes it always has real roots.
DERIVATIVES AND INTEGRALS
H\int \frac{t\, dt}{(\alpha t^{2} + \beta)\sqrtp{\gamma t^{2} + \delta}} + K\int \frac{dt}{(\alpha t^{2} + \beta)\sqrtp{\gamma t^{2} + \delta}}
After the substitution, integral (5) splits into H times one integral plus K times a second integral; this is equation (6).
DERIVATIVES AND INTEGRALS
\frac{t}{\sqrtp{\gamma t^{2} + \delta}} = u
Substitution that rationalises the second integral of equation (6).
DERIVATIVES AND INTEGRALS
\int \frac{dt}{(\alpha t^{2} + \beta) \sqrtp{\gamma t^{2} + \delta}} = \int \frac{du}{\beta + (\alpha\delta - \beta\gamma) u^{2}}
Under the substitution t/sqrt(gamma t^2 + delta) = u, the second integral in (6) becomes a rational integral in u.
DERIVATIVES AND INTEGRALS
\cos x = \frac{1 - t^{2}}{1 + t^{2}}
cos x expressed in terms of t, where t = tan(x/2).
DERIVATIVES AND INTEGRALS
\sin x = \frac{2t}{1 + t^{2}}
sin x expressed in terms of t, where t = tan(x/2).
DERIVATIVES AND INTEGRALS
\frac{dx}{dt} = \frac{2}{1 + t^{2}}
The derivative of x with respect to t = tan(x/2), so that the substitution reduces the integral to a rational function of t.
DERIVATIVES AND INTEGRALS
\int \phi(y)\, dy = \int xf'(x)\, dx = xf(x) - \int f(x)\, dx
If y = f(x) and phi is the inverse of f, the integral of phi(y) equals x f(x) minus the integral of f(x).
DERIVATIVES AND INTEGRALS
\int x^{m}(\log x)^{n}\, dx = \frac{x^{m+1} (\log x)^{n}}{m + 1} - \frac{n}{m + 1} \int x^{m}(\log x)^{n-1}\, dx
Integration by parts reduces the power n of log x by one at each step, so the integral can be completed by repetition.
DERIVATIVES AND INTEGRALS
(PRP') + (NN'RP) = (NN'P'P)
Additivity of areas: the area PRP' plus the area NN'RP equals the area NN'P'P, taken as one of the common-sense properties of area assumed in the text.
DERIVATIVES AND INTEGRALS
\Phi(x + h) - \Phi(x) = h\{\phi(x) + \mu(h)\}
The increase of the area function over an interval of length h equals h times the ordinate at x plus a small error mu(h).
DERIVATIVES AND INTEGRALS
|\mu(h)| < \lambda(h)
The error mu(h) is bounded in absolute value by lambda(h), the greatest distance of any point of the arc from the chord line PR, and lambda(h) tends to 0 as h tends to 0.
DERIVATIVES AND INTEGRALS
\Phi'(x) = \lim_{h \to 0} \frac{\Phi(x + h) - \Phi(x)}{h} = \lim_{h \to 0} \{\phi(x) + \mu(h)\} = \phi(x)
The derivative of the area function is the ordinate: the ordinate of the curve is the derivative of the area, so the area is the integral of the ordinate.
DERIVATIVES AND INTEGRALS
\{S(x + h) - S(x)\}/h = \{PP'\}/h = (PP'/h) × (\{PP'\}/PP')
The increment of the arc length S over h, divided by h, equals the chord ratio PP'/h times the ratio of the arc to its chord.
DERIVATIVES AND INTEGRALS
PP' + \sqrtp{PR^{2} + RP'^{2}} = h\bigsqrtp{1 + \frac{k^{2}}{h^{2}}}
As printed: a Pythagorean relation for the chord PP' with legs PR = h and RP' = k. The printed '+' between PP' and the root does not read as a standard identity and is flagged here as a possible transcription or printing error for '=', not corrected.
DERIVATIVES AND INTEGRALS
k = \phi(x + h) - \phi(x) = h\phi'(\xi)
The increment k of the ordinate equals h times the derivative at some point xi between x and x + h.
DERIVATIVES AND INTEGRALS
\lim (PP'/h) = \lim \sqrtb{1 + [\phi'(\xi)]^{2}} = \sqrtb{1 + [\phi'(x)]^{2}}
As h tends to 0 the ratio PP'/h tends to the square root of one plus the square of the derivative of phi at x.
DERIVATIVES AND INTEGRALS
\lim \{PP'\}/PP' = 1
Assumption that the arc PP' becomes indistinguishable from its chord as h tends to 0 (a hypothesis the text adopts, not proves).
DERIVATIVES AND INTEGRALS
S'(x) = \lim \{S(x + h) - S(x)\}/h = \sqrtb{1 + [\phi'(x)]^{2}}
The derivative of the arc length with respect to x equals the square root of one plus the square of the derivative of the ordinate.
DERIVATIVES AND INTEGRALS
S(x) = \int \sqrtb{1 + [\phi'(x)]^{2}}\, dx
The arc length of the curve y = phi(x) from the origin is the integral of the square root of one plus the square of phi'(x).
DERIVATIVES AND INTEGRALS
u_{0}^{2} u_{3} - 3u_{0} u_{1} u_{2} + 2u_{1}^{3}
A combination of the u_r, where u_r are the successive derivative-type functions a, ax+b, ax^2+2bx+c, ..., which the exercise shows is independent of x.
DERIVATIVES AND INTEGRALS
u_{0} u_{4} - 4u_{1} u_{3} + 3u_{2}^{2}
A second combination of the u_r which the exercise shows is independent of x.
DERIVATIVES AND INTEGRALS
U_{0}U_{2n} - 2nU_{1}U_{2n-1} + \frac{2n(2n - 1)}{1·2} U_{2}U_{2n-2} - \dots + U_{2n}U_{0}
An alternating binomial-weighted sum of products of the U_r which is independent of x.
DERIVATIVES AND INTEGRALS
U_{r}' = rU_{r-1}
The derivative of U_r with respect to x equals r times U_{r-1}.
DERIVATIVES AND INTEGRALS
y^{3} + 3yx + 2x^{3} = 0
The relation between x and y from which the second-derivative identity of Ex. 7 is deduced.
DERIVATIVES AND INTEGRALS
x^{2}(1 + x^{3})y'' - \frac{3}{2}xy' + y = 0
Second-order differential equation satisfied by y when y^3 + 3yx + 2x^3 = 0.
DERIVATIVES AND INTEGRALS
y = \phi\{\psi(y_{1})\} + \phi\{x - \psi(y_{1})\}
Differential equation of Ex. 8, where y_1 is the derivative of y and psi inverts phi', which the functions y = phi(c) + phi(x - c) and y = 2 phi(x/2) satisfy.
DERIVATIVES AND INTEGRALS
y = \phi(c) + \phi(x - c)
A solution of the differential equation of Ex. 8, with c a constant.
DERIVATIVES AND INTEGRALS
y = 2\phi(\frac{1}{2}x)
A second solution of the differential equation of Ex. 8.
DERIVATIVES AND INTEGRALS
y = \{x/\psi(y_{1})\} \phi\{\psi(y_{1})\}
Differential equation of Ex. 9, whose solutions include y = c phi(x/c) and y = beta x.
DERIVATIVES AND INTEGRALS
y = c\phi(x/c)
A solution of the differential equation of Ex. 9, with c a constant.
DERIVATIVES AND INTEGRALS
y = \beta x
A straight-line solution of the differential equation of Ex. 9.
DERIVATIVES AND INTEGRALS
\beta = \phi(\alpha)/\alpha
The constant beta is defined as phi(alpha) divided by alpha, where alpha is a root of phi(alpha) - alpha phi'(alpha) = 0.
DERIVATIVES AND INTEGRALS
\phi(\alpha) - \alpha\phi'(\alpha) = 0
The equation whose root alpha determines the constant beta in the solution of Ex. 9.
DERIVATIVES AND INTEGRALS
y_{2} = 0
The general differential equation of all straight lines ax + by + c = 0, with y_2 the second derivative of y with respect to x.
DERIVATIVES AND INTEGRALS
1 + y_{1}^{2} + yy_{2} = 0
The general differential equation of all circles with centres on the axis of x.
DERIVATIVES AND INTEGRALS
y_{1}^{2} + yy_{2} = 0
The general differential equation of all parabolas with axes along the axis of x.
DERIVATIVES AND INTEGRALS
y_{3} = 0
The general differential equation of all parabolas with axes parallel to the axis of y.
DERIVATIVES AND INTEGRALS
(1 + y_{1}^{2}) y_{3} = 3y_{1} y_{2}^{2}
The general differential equation of all circles.
DERIVATIVES AND INTEGRALS
5y_{3}^{2} = 3y_{2} y_{4}
The general differential equation of all parabolas.
DERIVATIVES AND INTEGRALS
9y_{2}^{2} y_{5} - 45y_{2} y_{3} y_{4} + 40y_{3}^{3} = 0
The general differential equation of all conics.
DERIVATIVES AND INTEGRALS
D_{x}^{2} (y_{2}^{-2/3}) = 0
The general differential equation of all parabolas.
DERIVATIVES AND INTEGRALS
D_{x}^{3} (y_{2}^{-2/3}) = 0
The general differential equation of all conics.
DERIVATIVES AND INTEGRALS
y_{2} = ±(pr - q^{2})/(px^{2} + 2qx + r)^{3/2}
The second derivative of a conic written as y = ax + b ± sqrt(px^2 + 2qx + r).
DERIVATIVES AND INTEGRALS
4ac - 5b^{2} = (4\alpha\gamma - 5\beta^{2})/\tau^{8}
Transformation relation between the coefficients a, b, c built from successive derivatives t, a, b, c and the reciprocal-derivative coefficients alpha, beta, gamma, tau.
DERIVATIVES AND INTEGRALS
bt - a^{2} = - (\beta\tau - \alpha^{2})/\tau^{6}
Second transformation identity relating the derivative coefficients in x and in y.
DERIVATIVES AND INTEGRALS
(1 - x^{2})y_{k+2} - (2k + 1)xy_{k+1} + (n^{2} - k^{2})y_{k} = 0
Recurrence for the k-th derivatives of y = sin(n arcsin x).
DERIVATIVES AND INTEGRALS
vD_{x}^{n}u = D_{x}^{n}(uv) - nD_{x}^{n-1}(uD_{x}v)
The first two terms of the generalised Leibniz-type formula expressing v D_x^n u in terms of derivatives of the product uv, for positive integer n.
DERIVATIVES AND INTEGRALS
x = a(2\cos t + \cos 2t)
Parametric equation of the curve of Ex. 15, x in terms of the parameter t.
DERIVATIVES AND INTEGRALS
y = a(2\sin t - \sin 2t)
Parametric equation of the curve of Ex. 15, y in terms of the parameter t.
DERIVATIVES AND INTEGRALS
x\sin \tfrac{1}{2} t + y\cos \tfrac{1}{2} t = a\sin \tfrac{3}{2} t
Equation of the tangent at the point with parameter t on the curve of Ex. 15.
DERIVATIVES AND INTEGRALS
x\cos \tfrac{1}{2} t - y\sin \tfrac{1}{2} t = 3a\cos \tfrac{3}{2} t
Equation of the normal at the point with parameter t on the curve of Ex. 15.
DERIVATIVES AND INTEGRALS
QR = 4a
The distance between the points Q and R where the tangent at P meets the curve is 4a.
This equation is in FUNCTIONS OF REAL VARIABLES (FUNCTIONS OF REAL VARIABLES)
DERIVATIVES AND INTEGRALS
x^{2} + y^{2} = 9a^{2}
The circle on which the normals at P, Q and R intersect.
DERIVATIVES AND INTEGRALS
(x^{2} + y^{2} + 12ax + 9a^{2})^{2} = 4a(2x + 3a)^{3}
The Cartesian equation of the curve parametrised in Ex. 15.
DERIVATIVES AND INTEGRALS
u^{2}\xi - u\eta = a(u^{3} - 1)
Equation of the tangent at the point defined by u (Ex. 16), in complex coordinates xi = x + yi and eta = x - yi.
DERIVATIVES AND INTEGRALS
u^{2}\xi + u\eta = 3a(u^{3} + 1)
Equation of the normal at the point defined by u (Ex. 16), in complex coordinates.
DERIVATIVES AND INTEGRALS
(p + q)^{2/3} - (p - q)^{2/3} = 1
The condition that x^4 + 4px^3 - 4qx - 1 = 0 should have equal roots.
DERIVATIVES AND INTEGRALS
\begin{vmatrix} f(a) & \phi(a) & \psi(a)\\ f(b) & \phi(b) & \psi(b)\\ f'(\xi) & \phi'(\xi) & \psi'(\xi) \end{vmatrix} =0
Generalised mean value theorem: a determinant built from f, phi, psi at a and b and their derivatives at some xi between a and b vanishes.
DERIVATIVES AND INTEGRALS
\frac{f(b) - f(a)}{\phi(b) - \phi(a)} = \frac{f'(\xi)}{\phi'(\xi)}\Add{.}
Cauchy-type quotient form of the mean value theorem deduced from Ex. 32, for some xi between a and b.
DERIVATIVES AND INTEGRALS
\phi(x) - \phi(x_{0}) = (x - x_{0})\phi'(\xi)
Mean value formula with x_0 < xi < x, used to prove the limit results of Ex. 34.
DERIVATIVES AND INTEGRALS
\phi(x) = 1/(1 + x^{2})
Definition of the function phi used in Ex. 31.
DERIVATIVES AND INTEGRALS
\phi^{n} (x) = Q_{n}(x)/(1 + x^{2})^{n+1}
The n-th derivative of 1/(1+x^2) has the form Q_n(x) over (1+x^2)^(n+1), with Q_n a polynomial of degree n.
DERIVATIVES AND INTEGRALS
Q_{n} = (-1)^{n} n!\left\{(n + 1)x^{n} - \dfrac{(n + 1)n(n - 1)}{3!} x^{n-2} + \dots\right\}
Leading terms of the polynomial Q_n in Ex. 31(iv).
DERIVATIVES AND INTEGRALS
\lambda(ax^{2} + bx + c) + \mu(a'x^{2} + b'x + c') = 0
The combined quadratic whose roots, by choice of the ratio lambda:mu, can be made real with any difference, unless the roots of the two quadratics interlace.
DERIVATIVES AND INTEGRALS
\pi < \frac{\sin \pi x}{x(1 - x)} \leq 4
Bounds on sin(pi x)/(x(1-x)) for 0 < x < 1.
DERIVATIVES AND INTEGRALS
\frac{dy}{dx} = \frac{(6x^{2} + x - 1) (x - 1)^{2} (x + 1)^{3}}{x^{2}}
The derivative of y with respect to x, whose sign analysis gives the general graph form in Ex. 23.
DERIVATIVES AND INTEGRALS
\arctan\{(a^{2} - b^{2})/2ab\}
The greatest acute angle at which the ellipse can be cut by a concentric circle.
DERIVATIVES AND INTEGRALS
s(x - s) x^{2} + 4\Delta^{2} = 0
Equation whose roots are the stationary values of one side of a triangle with fixed area and semi-perimeter.
DERIVATIVES AND INTEGRALS
s(s - a)(s - b)(s - c) = \Delta^{2}
Heron's formula in the form used in Ex. 26, relating area Delta to the semi-perimeter s and sides a, b, c.
DERIVATIVES AND INTEGRALS
a + b + c = 2s
The sum of the sides of a triangle is twice its semi-perimeter.
DERIVATIVES AND INTEGRALS
2\Delta + \frac{a^{2} + b^{2} + c^{2}}{2\sqrt{3}}
The area of the greatest equilateral triangle with sides through three given points A, B, C.
DERIVATIVES AND INTEGRALS
256\Delta\Delta' = 25a^{4}\sqrt{5}
Relation between the areas of the two maximum isosceles triangles on the cardioid r = a(1 + cos theta).
DERIVATIVES AND INTEGRALS
x^{2}y - 4x^{2} - 4xy + y^{2} + 16x - 2y - 7 = 0
The curve on which the point (x, y) approaches (2, 3) in Ex. 29.
DERIVATIVES AND INTEGRALS
(x^{2} - 4y + 8)/(y^{2} - 6x + 3)
The function whose limiting values as (x, y) approaches (2, 3) on the curve are found in Ex. 29.
DERIVATIVES AND INTEGRALS
\frac{d}{da}\{\lim_{x \to a} f(x)\} - \lim_{x \to a}f'(x) = \tfrac{3}{4} \sec^{3} a - \tfrac{5}{12} \sec a
The difference between the derivative in a of the limit of f and the limit of f' equals the stated secant expression.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{(1 + x^{2})^{3}}
Integral to be evaluated in Ex. 38 (a calculation exercise, listed only as an integral to compute).
DERIVATIVES AND INTEGRALS
2(n - 1)(q - \tfrac{1}{4}p^{2}) \int \frac{dx}{(x^{2} + px + q)^{n}} \\ = \frac{x + \frac{1}{2}p}{(x^{2} + px + q)^{n-1}} + (2n - 3) \int \frac{dx}{(x^{2} + px + q)^{n-1}}
Reduction formula expressing the integral of (x^2+px+q)^(-n) in terms of the integral with exponent n-1.
DERIVATIVES AND INTEGRALS
(p + 1) I_{p, q} = x^{p+1}(1 + x)^{q} - qI_{p+1, q-1}
Reduction formula for I_{p,q} = integral of x^p (1+x)^q dx.
DERIVATIVES AND INTEGRALS
I_{p, q} = (-1)^{p+1} \int y^{p} (1 + y)^{-p-q-2}\, dy
Result of the substitution x = -y/(1+y) applied to I_{p,q}.
DERIVATIVES AND INTEGRALS
\int xX^{-1/3}\, dx = -3(3a - 2bx) X^{2/3}/10b^{2}
Integral of x X^(-1/3) with X = a + bx.
DERIVATIVES AND INTEGRALS
\int x^{2}X^{-1/3}\, dx = 3(9a^{2} - 6abx + 5b^{2}x^{2}) X^{2/3}/40b^{3}
Integral of x^2 X^(-1/3) with X = a + bx.
DERIVATIVES AND INTEGRALS
2(n - 1)I_{m, n} = -x^{m-1} (1 + x^{2})^{-(n-1)} + (m - 1)I_{m-2, n-1}
Reduction formula for I_{m,n} = integral of x^m/(1+x^2)^n dx.
DERIVATIVES AND INTEGRALS
\beta I_{n} = x^{n} \sin\beta x - nJ_{n-1}
Reduction formula for I_n = integral of x^n cos(beta x) dx, linking it to J_{n-1}.
DERIVATIVES AND INTEGRALS
\beta J_{n} = -x^{n} \cos\beta x + nI_{n-1}
Reduction formula for J_n = integral of x^n sin(beta x) dx.
DERIVATIVES AND INTEGRALS
nI_{n} = \sin x\cos^{n-1} x + (n - 1) I_{n-2}
Reduction formula for I_n = integral of cos^n x dx.
DERIVATIVES AND INTEGRALS
nJ_{n} = -\cos x\sin^{n-1} x + (n - 1) J_{n-2}
Reduction formula for J_n = integral of sin^n x dx.
DERIVATIVES AND INTEGRALS
(n - 1)(I_{n} + I_{n-2}) = \tan^{n-1}x
Reduction formula for I_n = integral of tan^n x dx.
DERIVATIVES AND INTEGRALS
(m+n)I_{m, n} = -\cos^{m+1}x \sin^{n-1}x + (n - 1) I_{m, n-2}
First reduction formula for I_{m,n} = integral of cos^m x sin^n x dx.
DERIVATIVES AND INTEGRALS
(n - 1)(n - 2)I_{m, n} = (n - 2)^{2}I_{m, n-2} + m(m - 1)I_{m-2, n-2} \\ -x^{m-1} \cosec^{n-1}x \{m\sin x + (n - 2) x\cos x\}
Reduction formula for I_{m,n} = integral of x^m cosec^n x dx.
DERIVATIVES AND INTEGRALS
(n - 1)(a^{2} - b^{2}) I_{n} = -b\sin x (a + b\cos x)^{-(n-1)} + (2n - 3)aI_{n-1} - (n - 2)I_{n-2}
Reduction formula for I_n = integral of (a + b cos x)^(-n) dx.
DERIVATIVES AND INTEGRALS
4n(n + 1)(ab - h^{2})I_{n+2} - 2n(2n + 1)(a + b)I_{n+1} + 4n^{2}I_{n} = -\frac{d^{2} I_{n}}{dx^{2}}
Reduction identity for I_n = integral of (a cos^2 x + 2h cos x sin x + b sin^2 x)^(-n) dx, involving the second derivative of I_n.
DERIVATIVES AND INTEGRALS
(m + 1)I_{m, n} = x^{m+1}(\log x)^{n} - nI_{m, n-1}
Reduction formula for I_{m,n} = integral of x^m (log x)^n dx.
DERIVATIVES AND INTEGRALS
x^{m+1} \left\{\frac{(\log x)^{n}}{m + 1} - \frac{n(\log x)^{n-1}}{(m + 1)^{2}} + \frac{n(n - 1)(\log x)^{n-2}}{(m + 1)^{3}} - \dots + \frac{(-1)^{n}n!}{(m + 1)^{n+1}}\right\}
Closed form of the integral of x^m (log x)^n dx for positive integer n.
DERIVATIVES AND INTEGRALS
\phi'' + a^{2}\phi = 0
Differential equation of simple harmonic type whose most general solution is A cos ax + B sin ax, or rho cos(ax + epsilon).
DERIVATIVES AND INTEGRALS
\phi'^{2} + a^{2}\phi^{2} = a^{2}b^{2}
First integral of phi'' + a^2 phi = 0, with b a constant.
DERIVATIVES AND INTEGRALS
y' + \omega z = 0
First equation of the linear system whose most general solution y, z is sought in Ex. 42.
DERIVATIVES AND INTEGRALS
z' - \omega y = 0
Second equation of the linear system in Ex. 42.
DERIVATIVES AND INTEGRALS
x = \cos\phi + \frac{\sin\alpha \sin\phi}{1 - \cos^{2}\alpha \sin^{2}\phi}
Parametric x-coordinate of the curve whose area is found in Ex. 43.
DERIVATIVES AND INTEGRALS
\frac{1}{2}\pi(1 + \sin\alpha)^{2}/\sin\alpha
Area enclosed by the curve of Ex. 43.
DERIVATIVES AND INTEGRALS
a^{2}(\beta - \cos\beta\sin\beta)
Area of either loop of the locus of the middle point of the chord in Ex. 44.
DERIVATIVES AND INTEGRALS
\pi(a^{2} + \frac{1}{2}b^{2})
Area enclosed by the locus of the foot of the perpendicular from A to a tangent of the circle in Ex. 46.
DERIVATIVES AND INTEGRALS
\int \frac{dx}{(lx + my + n)(hx + by + f)} = \alpha\log \frac{PT}{PT'} + \beta
Integral along a conic expressed as a logarithm of ratio of tangent perpendiculars plus a constant.
DERIVATIVES AND INTEGRALS
\alpha e + \gamma = 0
Condition under which the integral of (alpha cos x + beta sin x + gamma)/(1 - e cos x)^2 is a rational function of cos x and sin x.

Problems

Exercise LII

Exercise LII, problem 1a, p. 247
Integrate $x\sin x$, $x^{2}\cos x$, $x^{2}\cos^{2}x$, $x^{2}\sin^{2}x \sin^{2} 2x$, $x\sin^{2}x \cos^{4}x$, $x^{3}\sin^{3}\frac{1}{3}x$.
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Exercise LII, problem 1b, p. 247
Integrate $x\sin x$, $x^{2}\cos x$, $x^{2}\cos^{2}x$, $x^{2}\sin^{2}x \sin^{2} 2x$, $x\sin^{2}x \cos^{4}x$, $x^{3}\sin^{3}\frac{1}{3}x$.
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Exercise LII, problem 1c, p. 247
Integrate $x\sin x$, $x^{2}\cos x$, $x^{2}\cos^{2}x$, $x^{2}\sin^{2}x \sin^{2} 2x$, $x\sin^{2}x \cos^{4}x$, $x^{3}\sin^{3}\frac{1}{3}x$.
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Exercise LII, problem 1d, p. 247
Integrate $x\sin x$, $x^{2}\cos x$, $x^{2}\cos^{2}x$, $x^{2}\sin^{2}x \sin^{2} 2x$, $x\sin^{2}x \cos^{4}x$, $x^{3}\sin^{3}\frac{1}{3}x$.
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Exercise LII, problem 1e, p. 247
Integrate $x\sin x$, $x^{2}\cos x$, $x^{2}\cos^{2}x$, $x^{2}\sin^{2}x \sin^{2} 2x$, $x\sin^{2}x \cos^{4}x$, $x^{3}\sin^{3}\frac{1}{3}x$.
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Exercise LII, problem 1f, p. 247
Integrate $x\sin x$, $x^{2}\cos x$, $x^{2}\cos^{2}x$, $x^{2}\sin^{2}x \sin^{2} 2x$, $x\sin^{2}x \cos^{4}x$, $x^{3}\sin^{3}\frac{1}{3}x$.
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Exercise LII, problem 2, p. 247
Find polynomials $P$ and $Q$ such that (3x - 1)x + (1 - 2x)x dx = Px + Qx.
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Exercise LII, problem 3, p. 247
Prove that $\ds\int x^{n}\cos x\, dx = P_{n}\cos x + Q_{n}\sin x$, where P_n = nx^n-1 - n(n - 1)(n - 2) x^n-3 + …,0pt minus 3ptQ_n = x^n - n(n - 1) x^n-2 + ….
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Exercise LIII

Exercise LIII, problem 1a, p. 247
Prove that x dx = |x + x|,0pt minus 3ptx dx = |12x|.
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Exercise LIII, problem 1b, p. 247
Prove that x dx = |x + x|,0pt minus 3ptx dx = |12x|.
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Exercise LIII, problem 2a, p. 247
$\ds\int \tan x\, dx = -\log |\cos x|$, $\ds\int \cot x\, dx = \log |\sin x|$, $\ds\int\sec^{2} x\, dx = \tan x$, $\ds\int \cosec^{2} x\, dx = -\cot x$, $\ds\int \tan x\sec x\, dx = \sec x$, $\ds\int \cot x \cosec x\, dx = -\cosec x$.
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Exercise LIII, problem 2b, p. 247
$\ds\int \tan x\, dx = -\log |\cos x|$, $\ds\int \cot x\, dx = \log |\sin x|$, $\ds\int\sec^{2} x\, dx = \tan x$, $\ds\int \cosec^{2} x\, dx = -\cot x$, $\ds\int \tan x\sec x\, dx = \sec x$, $\ds\int \cot x \cosec x\, dx = -\cosec x$.
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Exercise LIII, problem 2c, p. 247
$\ds\int \tan x\, dx = -\log |\cos x|$, $\ds\int \cot x\, dx = \log |\sin x|$, $\ds\int\sec^{2} x\, dx = \tan x$, $\ds\int \cosec^{2} x\, dx = -\cot x$, $\ds\int \tan x\sec x\, dx = \sec x$, $\ds\int \cot x \cosec x\, dx = -\cosec x$.
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Exercise LIII, problem 2d, p. 247
$\ds\int \tan x\, dx = -\log |\cos x|$, $\ds\int \cot x\, dx = \log |\sin x|$, $\ds\int\sec^{2} x\, dx = \tan x$, $\ds\int \cosec^{2} x\, dx = -\cot x$, $\ds\int \tan x\sec x\, dx = \sec x$, $\ds\int \cot x \cosec x\, dx = -\cosec x$.
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Exercise LIII, problem 2e, p. 247
$\ds\int \tan x\, dx = -\log |\cos x|$, $\ds\int \cot x\, dx = \log |\sin x|$, $\ds\int\sec^{2} x\, dx = \tan x$, $\ds\int \cosec^{2} x\, dx = -\cot x$, $\ds\int \tan x\sec x\, dx = \sec x$, $\ds\int \cot x \cosec x\, dx = -\cosec x$.
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Exercise LIII, problem 2f, p. 247
$\ds\int \tan x\, dx = -\log |\cos x|$, $\ds\int \cot x\, dx = \log |\sin x|$, $\ds\int\sec^{2} x\, dx = \tan x$, $\ds\int \cosec^{2} x\, dx = -\cot x$, $\ds\int \tan x\sec x\, dx = \sec x$, $\ds\int \cot x \cosec x\, dx = -\cosec x$.
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Exercise LIII, problem 3, p. 247
Show that the integral of $1/(a + b\cos x)$, where $a + b$ is positive, may be expressed in one or other of the forms 2a^2 - b^2 ta - ba + b,0pt minus 3pt1b^2 - a^2 |b + a + tb - a b + a - tb - a|, where $t = \tan\frac{1}{2}x$, according as $a^{2} > b^{2}$ or $a^{2} < b^{2}$. If $a^{2} = b^{2}$ then the integral reduces to a constant multiple of that of $\sec^{2}\frac{1}{2}x$ or $\cosec^{2}\frac{1}{2}x$, and its value may at once be written down. Deduce the forms of the integral when $a + b$ is negative.
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Exercise LIII, problem 4, p. 247
Show that if $y$ is defined in terms of $x$ by means of the equation [ (a + bcos x)(a - bcos y) = a^2 - b^2, ] where $a$ is positive and $a^{2} > b^{2}$, then as $x$ varies from $0$ to $\\pi$ one value of $y$ also varies from $0$ to $\\pi$. Show also that [ sin x = fracsqrtpa^2 - b^2 sin ya - bcos y,quad fracsin xa + bcos x, fracdxdy = fracsin ya - bcos y; PageSep248 and deduce that if $0 < x < \\pi$ then [ int fracdxa + bcos x = frac1sqrtpa^2 - b^2 arccos left(fracacos x + ba + bcos xright). ] Show that this result agrees with that of Ex. 3.
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Exercise LIII, problem 5, p. 247
Show how to integrate $1/(a + b\cos x + c\sin x)$.
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Exercise LIII, problem 6, p. 247
Integrate $(a + b\cos x + c\sin x)/(\alpha + \beta\cos x + \gamma\sin x)$
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Exercise LIII, problem 7, p. 247
Integrate $1/(a\cos^{2} x + 2b\cos x\sin x + c\sin^{2} x)$.
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Exercise LIV

Exercise LIV, problem 1, p. 251
Calculate the area of the segment cut off from the parabola $y = x^{2}/4a$ by the ordinate $x = \xi$, and the length of the arc which bounds it.
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Exercise LIV, problem 10, p. 251
Find the area of the loop of the curve $x^{5} + y^{5} = 5ax^{2}y^{2}$.
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Exercise LIV, problem 11, p. 251
Prove that the area of a loop of the curve $x = a\sin 2t$, $y = a\sin t$ is $\frac{4}{3}a^{2}$. % [0]% (*Math. Trip.* 1908.)% [1]%
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Exercise LIV, problem 12, p. 251
The arc of the ellipse given by $x = a\cos t$, $y = b\sin t$, between the points $t = t_{1}$ and $t = t_{2}$, is $F(t_{2}) - F(t_{1})$, where F(t) = a1 - e^2^2 t dt, $e$ being the eccentricity. [This integral cannot however be evaluated in terms of such functions as are at present at our disposal.]
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Exercise LIV, problem 13a, p. 251
**coordinates.** Show that the area bounded by the curve $r = f(\theta)$, where $f(\theta)$ is a one-valued function of $\theta$, and the radii $\theta = \theta_{1}$, $\theta = \theta_{2}$, is $F(\theta_{2}) - F(\theta_{1})$, where $\ds F(\theta) = \tfrac{1}{2} \int r^{2}\, d\theta$. And the length of the corresponding arc of the curve is $\Phi(\theta_{2}) - \Phi(\theta_{1})$, where () = r^2 + (drd)^2 d. Hence determine (i) the area and perimeter of the circle $r = 2a\sin\theta$; (ii) the area between the parabola $r = \frac{1}{2}l\sec^{2} \frac{1}{2}\theta$ and its latus rectum, and the length of the corresponding arc of the parabola; (iii) the area of the limaçon $r = a + b\cos\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$; and (iv) the areas of the ellipses $1/r^{2} = a\cos^{2} \theta + 2h\cos\theta\sin\theta + b\sin^{2} \theta$ and $l/r = 1 + e\cos\theta$. [In the last case we are led to the integral $\ds \int \frac{d\theta}{(1 + e\cos\theta)^{2}}$, which may be calculated (cf. % [examples:liii]Ex. liii%. 4) by the help of the substitution (1 + e) (1 - e) = 1 - e^2.]
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Exercise LIV, problem 13b, p. 251
**coordinates.** Show that the area bounded by the curve $r = f(\theta)$, where $f(\theta)$ is a one-valued function of $\theta$, and the radii $\theta = \theta_{1}$, $\theta = \theta_{2}$, is $F(\theta_{2}) - F(\theta_{1})$, where $\ds F(\theta) = \tfrac{1}{2} \int r^{2}\, d\theta$. And the length of the corresponding arc of the curve is $\Phi(\theta_{2}) - \Phi(\theta_{1})$, where () = r^2 + (drd)^2 d. Hence determine (i) the area and perimeter of the circle $r = 2a\sin\theta$; (ii) the area between the parabola $r = \frac{1}{2}l\sec^{2} \frac{1}{2}\theta$ and its latus rectum, and the length of the corresponding arc of the parabola; (iii) the area of the limaçon $r = a + b\cos\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$; and (iv) the areas of the ellipses $1/r^{2} = a\cos^{2} \theta + 2h\cos\theta\sin\theta + b\sin^{2} \theta$ and $l/r = 1 + e\cos\theta$. [In the last case we are led to the integral $\ds \int \frac{d\theta}{(1 + e\cos\theta)^{2}}$, which may be calculated (cf. % [examples:liii]Ex. liii%. 4) by the help of the substitution (1 + e) (1 - e) = 1 - e^2.]
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Exercise LIV, problem 13c, p. 251
**coordinates.** Show that the area bounded by the curve $r = f(\theta)$, where $f(\theta)$ is a one-valued function of $\theta$, and the radii $\theta = \theta_{1}$, $\theta = \theta_{2}$, is $F(\theta_{2}) - F(\theta_{1})$, where $\ds F(\theta) = \tfrac{1}{2} \int r^{2}\, d\theta$. And the length of the corresponding arc of the curve is $\Phi(\theta_{2}) - \Phi(\theta_{1})$, where () = r^2 + (drd)^2 d. Hence determine (i) the area and perimeter of the circle $r = 2a\sin\theta$; (ii) the area between the parabola $r = \frac{1}{2}l\sec^{2} \frac{1}{2}\theta$ and its latus rectum, and the length of the corresponding arc of the parabola; (iii) the area of the limaçon $r = a + b\cos\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$; and (iv) the areas of the ellipses $1/r^{2} = a\cos^{2} \theta + 2h\cos\theta\sin\theta + b\sin^{2} \theta$ and $l/r = 1 + e\cos\theta$. [In the last case we are led to the integral $\ds \int \frac{d\theta}{(1 + e\cos\theta)^{2}}$, which may be calculated (cf. % [examples:liii]Ex. liii%. 4) by the help of the substitution (1 + e) (1 - e) = 1 - e^2.]
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Exercise LIV, problem 13d, p. 251
**coordinates.** Show that the area bounded by the curve $r = f(\theta)$, where $f(\theta)$ is a one-valued function of $\theta$, and the radii $\theta = \theta_{1}$, $\theta = \theta_{2}$, is $F(\theta_{2}) - F(\theta_{1})$, where $\ds F(\theta) = \tfrac{1}{2} \int r^{2}\, d\theta$. And the length of the corresponding arc of the curve is $\Phi(\theta_{2}) - \Phi(\theta_{1})$, where () = r^2 + (drd)^2 d. Hence determine (i) the area and perimeter of the circle $r = 2a\sin\theta$; (ii) the area between the parabola $r = \frac{1}{2}l\sec^{2} \frac{1}{2}\theta$ and its latus rectum, and the length of the corresponding arc of the parabola; (iii) the area of the limaçon $r = a + b\cos\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$; and (iv) the areas of the ellipses $1/r^{2} = a\cos^{2} \theta + 2h\cos\theta\sin\theta + b\sin^{2} \theta$ and $l/r = 1 + e\cos\theta$. [In the last case we are led to the integral $\ds \int \frac{d\theta}{(1 + e\cos\theta)^{2}}$, which may be calculated (cf. % [examples:liii]Ex. liii%. 4) by the help of the substitution (1 + e) (1 - e) = 1 - e^2.]
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Exercise LIV, problem 14, p. 251
Trace the curve $2\theta = (a/r) + (r/a)$, and show that the area bounded by the radius vector $\theta = \beta$, and the two branches which touch at the point $r = a$, $\theta = 1$, is $\frac{2}{3} a^{2}(\beta^{2} - 1)^{3/2}$. % [0]% (*Math. Trip.* 1900.)% [1]%
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Exercise LIV, problem 15, p. 251
A curve is given by an equation $p = f(r)$, $r$ being the radius vector and $p$ the perpendicular from the origin on to the tangent. Show that the calculation of the area of the region bounded by an arc of the curve and two radii vectores depends upon that of the integral $\frac{1}{2} \ds \int \frac{pr\, dr}{\sqrtp{r^{2} - p^{2}}}$.
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Exercise LIV, problem 2, p. 251
Answer the same questions for the curve $ay^{2} = x^{3}$, showing that the length of the arc is 8a27 (1 + 94a)^3/2 - 1.
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Exercise LIV, problem 3, p. 251
Calculate the areas and lengths of the circles $x^{2} + y^{2} = a^{2}$, $x^{2} + y^{2} = 2ax$ by means of the formulae of [§§]145--146.
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Exercise LIV, problem 4, p. 251
Show that the area of the ellipse $(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1$ is $\pi ab$.
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Exercise LIV, problem 5, p. 251
Find the area bounded by the curve $y = \sin x$ and the segment of the axis of $x$ from $x = 0$ to $x = 2\pi$. [Here $\Phi(x) = -\cos x$, and the difference between the values of $-\cos x$ for $x = 0$ and $x = 2\pi$ is zero. The explanation of this is of course that between $x = \pi$ and $x = 2\pi$ the curve lies below the axis of $x$, and so the corresponding part of the area is counted negative in applying the method. The area from $x = 0$ to $x = \pi$ is $-\cos \pi + \cos 0 = 2$; and the whole area required, when every part is counted positive, is twice this, *i.e.* is $4$.]
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Exercise LIV, problem 6, p. 251
Suppose that the coordinates of any point on a curve are expressed as functions of a parameter $t$ by equations of the type $x = \phi(t)$, $y = \psi(t)$, $\phi$ and $\psi$ being functions of $t$ with continuous derivatives. Prove that if $x$ steadily increases as $t$ varies from $t_{0}$ to $t_{1}$, then the area of the region bounded by the corresponding portion of the curve, the axis of $x$, and the two ordinates corresponding to $t_{0}$ and $t_{1}$, is, apart from sign, $A(t_{1}) - A(t_{0})$, where A(t) = (t)’(t) dt = y dxdt dt.
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Exercise LIV, problem 7, p. 251
Suppose that $C$ is a closed curve formed of a single loop and not met by any parallel to either axis in more than two points. And suppose that the coordinates of any point $P$ on the curve can be expressed as in Ex. 6 in terms of $t$, and that, as $t$ varies from $t_{0}$ to $t_{1}$, $P$ moves in the same direction round the curve and returns after a single circuit to its original position. Show that the area of the loop is equal to the difference of the initial and final values of any one of the integrals -y dxdt dt,0pt minus 3pt x dydt dt,0pt minus 3pt12 (x dydt - y dxdt) dt, this difference being of course taken positively.
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Exercise LIV, problem 8a, p. 251
Apply the result of Ex. 7 to determine the areas of the curves given by % [2.25em][l](i)% [2.25em][l](i)% % xa = 1 - t^21 + t^2,0pt minus 3ptya = 2t1 + t^2, % [2.25em][l](ii)% [2.25em][l](ii)% % x = a^3 t,0pt minus 3pty = b^3 t.
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Exercise LIV, problem 8b, p. 251
Apply the result of Ex. 7 to determine the areas of the curves given by % [2.25em][l](i)% [2.25em][l](i)% % xa = 1 - t^21 + t^2,0pt minus 3ptya = 2t1 + t^2, % [2.25em][l](ii)% [2.25em][l](ii)% % x = a^3 t,0pt minus 3pty = b^3 t.
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Exercise LIV, problem 9, p. 251
Find the area of the loop of the curve $x^{3} + y^{3} = 3axy$. [Putting $y = tx$ we obtain $x = 3at/(1 + t^{3})$, $y = 3at^{2}/(1 + t^{3})$. As $t$ varies from $0$ towards $\infty$ the loop is described once. Also 12 (y dxdt - x dydt) dt = -12 x^2 ddt(yx) dt = -12 9a^2t^2(1 + t^3)^2 dt = 3a^22(1 + t^3), which tends to $0$ as $t \to \infty$. Thus the area of the loop is $\frac{3}{2}a^{2}$.]
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Exercise XXXIX

Exercise XXXIX, problem 1, p. 201
If $\phi(x)$ is a constant then $\phi'(x) = 0$. Interpret this result geometrically.
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Exercise XXXIX, problem 2, p. 201
If $\phi(x) = ax + b$ then $\phi'(x) = a$. Prove this (i) from the formal definition and (ii) by geometrical considerations.
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Exercise XXXIX, problem 3, p. 201
If $\phi(x) = x^{m}$, where $m$ is a positive integer, then $\phi'(x) = mx^{m-1}$. [For align* ’(x) &= (x + h)^m - x^mh &= mx^m-1 + m(m - 1)1·2 x^m-2 h + …+ h^m-1. align* The reader should observe that this method cannot be applied to $x^{p/q}$, where $p/q$ is a rational fraction, as we have no means of expressing $(x + h)^{p/q}$ as a finite series of powers of $h$. We shall show later on ([§]118) that the result of this example holds for all rational values of $m$. Meanwhile the reader will find it instructive to determine $\phi'(x)$ when $m$ has some special fractional value (*e.g.* $\frac{1}{2}$), by means of some special device.]
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Exercise XXXIX, problem 4, p. 201
0.375em plus 0.75em minus 0.25emIf $\phi(x) = \sin x$, then $\phi'(x) = \cos x$; and if $\phi(x) = \cos x$, then $\phi'(x) = -\sin x$. [For example, if $\phi(x) = \sin x$, we have (x + h) - (x)/h = 212h (x + 12h)/h, the limit of which, when $h \to 0$, is $\cos x$, since $\lim\cos(x + \frac{1}{2}h) = \cos x$ (the cosine being a continuous function) and $\lim\{(\sin \frac{1}{2}h)/\frac{1}{2}h\} = 1$ (% [examples:xxxvi]Ex. xxxvi%. 13).]
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Exercise XXXIX, problem 5, p. 201
**of the tangent and normal to a curve $y = \phi(x)$.** The tangent to the curve at the point $(x_{0}, y_{0})$ is the line through $(x_{0}, y_{0})$ which makes with $OX$ an angle $\psi$, where $\tan\psi = \phi'(x_{0})$. Its equation is therefore y - y_0 = (x - x_0) ’(x_0); and the equation of the normal (the perpendicular to the tangent at the point of contact) is (y - y_0) ’(x_0) + x - x_0 = 0. We have assumed that the tangent is not parallel to the axis of $y$. In this special case it is obvious that the tangent and normal are $x = x_{0}$ and $y = y_{0}$ respectively.
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Exercise XXXIX, problem 6, p. 201
Write down the equations of the tangent and normal at any point of the parabola $x^{2} = 4ay$. Show that if $x_{0} = 2a/m$, $y_{0} = a/m^{2}$, then the tangent at $(x_{0}, y_{0})$ is $x = my + (a/m)$.
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Exercise XL

Exercise XL, problem 1, p. 206
If $y = y_{1}y_{2}y_{3}$ then dydx = y_2y_3 dy_1dx + y_3y_1 dy_2dx + y_1y_2 dy_3dx, and if $y = y_{1}y_{2} \dots y_{n}$ then dydx = _r=1^n y_1y_2 …y_r-1y_r+1 …y_n dy_rdx. In particular, if $y = z^{n}$, then $dy/dx = nz^{n-1}(dz/dx)$; and if $y = x^{n}$, then $dy/dx = nx^{n-1}$, as was proved otherwise in % [examples:xxxix]Ex. xxxix%. 3.
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Exercise XL, problem 2, p. 206
If $y = y_{1}y_{2}\dots y_{n}$ then 1y dydx = 1y_1 dy_1dx + 1y_2 dy_2dx + … + 1y_n dy_ndx. In particular, if $y = z^{n}$, then $\dfrac{1}{y}\, \dfrac{dy}{dx} = \dfrac{n}{z}\, \dfrac{dz}{dx}$.
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- If $y = y_{1}y_{2}\dots y_{n}$ then 1y dydx = 1y_1 dy_1dx + 1y_2 dy_2dx + … + 1y_n dy_ndx. In particular, if $y = z^{n}$, then $\dfrac{1}{y}\, \dfrac{dy}{dx} = \dfrac{n}{z}\, \dfrac{dz}{dx}$.
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Exercise XLI

Exercise XLI, problem 1, p. 208
Show that if $\phi(x)$ is a polynomial then $\phi'(x)$ is the coefficient of $h$ in the expansion of $\phi(x + h)$ in powers of $h$.
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Exercise XLI, problem 10, p. 208
**’s Theorem for polynomials.** If $\phi(x)$ is any polynomial, then between any pair of roots of $\phi(x) = 0$ lies a root of $\phi'(x) = 0$.
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Exercise XLI, problem 2, p. 208
If $\phi(x)$ is divisible by $(x - \alpha)^{2}$, then $\phi'(x)$ is divisible by $x - \alpha$: and generally, if $\phi(x)$ is divisible by $(x - \alpha)^{m}$, then $\phi'(x)$ is divisible by $(x - \alpha)^{m-1}$.
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Exercise XLI, problem 3, p. 208
Conversely, if $\phi(x)$ and $\phi'(x)$ are *both* divisible by $x - \alpha$, then $\phi(x)$ is divisible by $(x - \alpha)^{2}$; and if $\phi(x)$ is divisible by $x - \alpha$ and $\phi'(x)$ by $(x - \alpha)^{m-1}$, then $\phi(x)$ is divisible by $(x - \alpha)^{m}$.
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Exercise XLI, problem 4, p. 208
Show how to determine as completely as possible the multiple roots of $P(x) = 0$, where $P(x)$ is a polynomial, with their degrees of multiplicity, by means of the elementary algebraical operations. [If $H_{1}$ is the highest common factor of $P$ and $P'$, $H_{2}$ the highest common factor of $H_{1}$ and $P''$, $H_{3}$ that of $H_{2}$ and $P'''$, and so on, then the roots of $H_{1}H_{3}/H_{2}^{2} = 0$ are the *double* roots of $P = 0$, the roots of $H_{2}H_{4}/H_{3}^{2} = 0$ the *treble* roots, and so on. But it may not be possible to complete the solution of $H_{1}H_{3}/H_{2}^{2} = 0$, $H_{2}H_{4}/H_{3}^{2} = 0$, …. Thus if $P(x) = (x - 1)^{3}(x^{5} - x - 7)^{2}$ then $H_{1}H_{3}/H_{2}^{2} = x^{5} - x - 7$ and $H_{2}H_{4}/H_{3}^{2} = x - 1$; and we cannot solve the first equation.]
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Exercise XLI, problem 5a, p. 208
Find all the roots, with their degrees of multiplicity, of x^4 + 3x^3 - 3x^2 - 11x - 6 = 0,0pt minus 3ptx^6 + 2x^5 - 8x^4 - 14x^3 + 11x^2 + 28x + 12 = 0.
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Exercise XLI, problem 5b, p. 208
Find all the roots, with their degrees of multiplicity, of x^4 + 3x^3 - 3x^2 - 11x - 6 = 0,0pt minus 3ptx^6 + 2x^5 - 8x^4 - 14x^3 + 11x^2 + 28x + 12 = 0.
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Exercise XLI, problem 6, p. 208
If $ax^{2} + 2bx + c$ has a double root, *i.e.* is of the form $a(x - \alpha)^{2}$, then $2(ax + b)$ must be divisible by $x - \alpha$, so that $\alpha = -b/a$. This value of $x$ must satisfy $ax^{2} + 2bx + c = 0$. Verify that the condition thus arrived at is $ac - b^{2} = 0$.
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Exercise XLI, problem 7, p. 208
The equation $1/(x - a) + 1/(x - b) + 1/(x - c) = 0$ can have a pair of equal roots only if $a = b = c$. % [0]% (*Math. Trip.* 1905.)% [1]%
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Exercise XLI, problem 8, p. 208
Show that ax^3 + 3bx^2 + 3cx + d = 0 has a double root if $G^{2} + 4H^{3} = 0$, where $H = ac - b^{2}$, $G = a^{2}d - 3abc + 2b^{3}$. [Put $ax + b = y$, when the equation reduces to $y^{3} + 3Hy + G = 0$. This must have a root in common with $y^{2} + H = 0$.]
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Exercise XLI, problem 9, p. 208
The reader may verify that if $\alpha$, $\beta$, $\gamma$, $\delta$ are the roots of ax^4 + 4bx^3 + 6cx^2 + 4dx + e = 0, then the equation whose roots are 112a (- )(- ) - (- )(- ) , and two similar expressions formed by permuting $\alpha$, $\beta$, $\gamma$ cyclically, is 4^3 - g_2- g_3 = 0, where g_2 = ae - 4bd + 3c^2,0pt minus 3ptg_3 = ace + 2bcd - ad^2 - eb^2 - c^3. It is clear that if two of $\alpha$, $\beta$, $\gamma$, $\delta$ are equal then two of the roots of this cubic will be equal. Using the result of Ex. 8 we deduce that $g_{2}^{3} - 27g_{3}^{2} = 0$.
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Exercise XLII

Exercise XLII, problem 1, p. 210
Prove that ddx(x1 + x^2) = 1 - x^2(1 + x^2)^2,0pt minus 3ptddx(1 - x^21 + x^2) = -4x(1 + x^2)^2.
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- Prove that ddx(x1 + x^2) = 1 - x^2(1 + x^2)^2,0pt minus 3ptddx(1 - x^21 + x^2) = -4x(1 + x^2)^2.
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Exercise XLII, problem 2, p. 210
Prove that ddx(ax^2 + 2bx + cAx^2 + 2Bx + C) = (ax + b) (Bx + C) - (bx + c) (Ax + B)(Ax^2 + 2Bx + C)^2.
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- Prove that ddx(ax^2 + 2bx + cAx^2 + 2Bx + C) = (ax + b) (Bx + C) - (bx + c) (Ax + B)(Ax^2 + 2Bx + C)^2.
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Exercise XLII, problem 3, p. 210
If $Q$ has a factor $(x - \alpha)^{m}$ then the denominator of $R'$ (when $R'$ is reduced to its lowest terms) is divisible by $(x - \alpha)^{m+1}$ but by no higher power of $x - \alpha$.
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- If $Q$ has a factor $(x - \alpha)^{m}$ then the denominator of $R'$ (when $R'$ is reduced to its lowest terms) is divisible by $(x - \alpha)^{m+1}$ but by no higher power of $x - \alpha$.
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Exercise XLII, problem 4, p. 210
In no case can the denominator of $R'$ have a *simple* factor $x - \alpha$. Hence no rational function (such as $1/x$) whose denominator contains any simple factor can be the derivative of another rational function.
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- In no case can the denominator of $R'$ have a *simple* factor $x - \alpha$. Hence no rational function (such as $1/x$) whose denominator contains any simple factor can be the derivative of another rational function.
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Exercise XLIII

Exercise XLIII, problem 1a, p. 211
Find the derivatives of 1 + x1 - x,0pt minus 3ptax + bcx + d,0pt minus 3ptax^2 + 2bx + cAx^2 + 2Bx + C,0pt minus 3pt(ax + b)^m (cx + d)^n.
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Exercise XLIII, problem 1b, p. 211
Find the derivatives of 1 + x1 - x,0pt minus 3ptax + bcx + d,0pt minus 3ptax^2 + 2bx + cAx^2 + 2Bx + C,0pt minus 3pt(ax + b)^m (cx + d)^n.
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Exercise XLIII, problem 1c, p. 211
Find the derivatives of 1 + x1 - x,0pt minus 3ptax + bcx + d,0pt minus 3ptax^2 + 2bx + cAx^2 + 2Bx + C,0pt minus 3pt(ax + b)^m (cx + d)^n.
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Exercise XLIII, problem 1d, p. 211
Find the derivatives of 1 + x1 - x,0pt minus 3ptax + bcx + d,0pt minus 3ptax^2 + 2bx + cAx^2 + 2Bx + C,0pt minus 3pt(ax + b)^m (cx + d)^n.
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Exercise XLIII, problem 2a, p. 211
Prove that ddxxa^2 + x^2 = a^2(a^2 + x^2)^(3/2),0pt minus 3ptddxxa^2 - x^2 = a^2(a^2 - x^2)^3/2.
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Exercise XLIII, problem 2b, p. 211
Prove that ddxxa^2 + x^2 = a^2(a^2 + x^2)^(3/2),0pt minus 3ptddxxa^2 - x^2 = a^2(a^2 - x^2)^3/2.
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Exercise XLIII, problem 3i, p. 211
Find the differential coefficient of $y$ when % [2.25em][l](i)% [2.25em][l](i)% % ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0,0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % x^5 + y^5 - 5ax^2y^2 = 0.
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Exercise XLIII, problem 3ii, p. 211
Find the differential coefficient of $y$ when % [2.25em][l](i)% [2.25em][l](i)% % ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0,0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % x^5 + y^5 - 5ax^2y^2 = 0.
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Exercise XLIV

Exercise XLIV, problem 1, p. 212
Find the derivatives of In these examples $m$ is a rational number and $a$, $b$, …, $\alpha$, $\beta$ … have such values that the functions which involve them are real. gather* ^m x, 0pt minus 3pt^m x, 0pt minus 3ptx^m, 0pt minus 3ptx^m, 0pt minus 3pt(x), 0pt minus 3pt(x), a^2^2 x + b^2^2 x, 0pt minus 3ptxxa^2^2 x + b^2^2 x, xx + 1 - x^2, 0pt minus 3pt(1 + x)x - x. gather*
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Exercise XLIV, problem 10, p. 212
Prove that the derivative of $F[f\{\phi(x)\}]$ is $F'[f\{\phi(x)\}]\, f'\{\phi(x)\}\phi'(x)$, and extend the result to still more complicated cases.
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Exercise XLIV, problem 11, p. 212
If $u$ and $v$ are functions of $x$, then D_x (u/v) = (vD_xu - uD_xv)/(u^2 + v^2).
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Exercise XLIV, problem 12, p. 212
The derivative of $y = (\tan x + \sec x)^{m}$ is $my\sec x$.
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Exercise XLIV, problem 13, p. 212
The derivative of $y = \cos x + i\sin x$ is $iy$.
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Exercise XLIV, problem 14, p. 212
Differentiate $x\cos x$, $(\sin x)/x$. Show that the values of $x$ for which the tangents to the curves $y = x\cos x$, $y = (\sin x)/x$ are parallel to the axis of $x$ are roots of $\cot x = x$, $\tan x = x$ respectively.
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Exercise XLIV, problem 15, p. 212
It is easy to see (cf. % [examples:xvii]Ex. xvii%. 5) that the equation $\sin x = ax$, where $a$ is positive, has no real roots except $x = 0$ if $a \geq 1$, and if $a < 1$ a finite number of roots which increases as $a$ diminishes. Prove that the values of $a$ for which the number of roots changes are the values of $\cos\xi$, where $\xi$ is a positive root of the equation $\tan\xi = \xi$. [The values required are the values of $a$ for which $y = ax$ touches $y = \sin x$.]
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How it was checked
- other: not a kind the checker handles
Exercise XLIV, problem 16, p. 212
If $\phi(x) = x^{2}\sin(1/x)$ when $x \neq 0$, and $\phi(0) = 0$, then ’(x) = 2x(1/x) - (1/x) when $x\neq 0$, and $\phi'(0) = 0$. And $\phi'(x)$ is discontinuous for $x = 0$ (cf. [§]111, (2)).
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- other: not a kind the checker handles
Exercise XLIV, problem 17, p. 212
Find the equations of the tangent and normal at the point $(x_{0}, y_{0})$ of the circle $x^{2} + y^{2} = a^{2}$. [Here $y = \sqrtp{a^{2} - x^{2}}$, $dy/dx = -x/\sqrtp{a^{2} - x^{2}}$, and the tangent is y - y_0 = (x - x_0) -x_0/a^2 - x_0^2, which may be reduced to the form $xx_{0} + yy_{0} = a^{2}$. The normal is $xy_{0} - yx_{0} = 0$, which of course passes through the origin.]
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- other: not a kind the checker handles
Exercise XLIV, problem 18, p. 212
Find the equations of the tangent and normal at any point of the ellipse $(x/a)^{2} + (y/b)^{2} = 1$ and the hyperbola $(x/a)^{2} - (y/b)^{2} = 1$.
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- other: not a kind the checker handles
Exercise XLIV, problem 19, p. 212
The equations of the tangent and normal to the curve $x = \phi(t)$, $y = \psi(t)$, at the point whose parameter is $t$, are x - (t)’(t) = y - (t)’(t),0pt minus 3ptx - (t) ’(t) + y - (t) ’(t) = 0.
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- (none printed)
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- other: not a kind the checker handles
Exercise XLIV, problem 2, p. 212
Verify by differentiation that $\arcsin x + \arccos x$ is constant for all values of $x$ between $0$ and $1$, and $\arctan x + \arccot x$ for all positive values of $x$.
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- other: not a kind the checker handles
Exercise XLIV, problem 3, p. 212
1 - x^2,0pt minus 3pt2x1 - x^2,0pt minus 3pt(a + x1 - ax). How do you explain the simplicity of the results?
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- (none printed)
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How it was checked
- differentiate: no printed answer to check
Exercise XLIV, problem 4, p. 212
1ac - b^2 ax + bac - b^2,0pt minus 3pt-1-a ax + bb^2 - ac.
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- (none printed)
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- differentiate: no printed answer to check
Exercise XLIV, problem 5, p. 212
Show that each of the functions 2x - - ,0pt minus 3pt2x - - x,0pt minus 3pt2(- x)(x - )- has the derivative 1(- x)(x - ).
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- other: not a kind the checker handles
Exercise XLIV, problem 6, p. 212
Prove that dd 3^3 = 33. % [0]% (*Math. Trip.* 1904.)% [1]%
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Exercise XLIV, problem 7, p. 212
Show that 1C(Ac - aC) ddx [ C(ax^2 + c)c(Ax^2 + C) ] = 1(Ax^2 + C) ax^2 + c.
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Exercise XLIV, problem 8, p. 212
Each of the functions 1a^2 - b^2 (ax + ba + bx),0pt minus 3pt2a^2 - b^2 a - ba + b 12x has the derivative $1/(a + b\cos x)$.
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Exercise XLIV, problem 9, p. 212
If $X = a + b\cos x + c\sin x$, and y = 1a^2 - b^2 -c^2 aX - a^2 + b^2 + c^2X b^2 + c^2, then $dy/dx = 1/X$.
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Exercise XLVII

Exercise XLVII, problem 1, p. 227
Show that (b) - (x) - b - xb - a(b) - (a) is the difference between the ordinates of a point on the curve and the corresponding point on the chord.
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Exercise XLVII, problem 2, p. 227
Verify the theorem when $\phi(x) = x^{2}$ and when $\phi(x) = x^{3}$.
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Exercise XLVII, problem 3, p. 227
Establish the theorem stated at the end of [§]124 by means of the Mean Value Theorem.
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Exercise XLVII, problem 4, p. 227
Use the Mean Value Theorem to prove Theorem (6) of [§]113, assuming that the derivatives which occur are continuous.
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Exercise XLVIII

Exercise XLVIII, problem 1, p. 235
Prove that Ax + Bax^2 + 2bx + c dx = A2a |X| + D2a - |ax + b - -ax + b + -| (where $X = ax^{2} + bx + c$) if $\Delta < 0$, and Ax + Bax^2 + 2bx + c dx = A2a |X| + D2a (ax + b) if $\Delta > 0$, $\Delta$ and $D$ having the same meanings as on p.234.
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Exercise XLVIII, problem 2, p. 235
In the particular case in which $ac = b^{2}$ the integral is -Da(ax + b) + Aa |ax + b|.
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Exercise XLVIII, problem 3, p. 235
Show that if the roots of $Q(x) = 0$ are all real and distinct, and $P(x)$ is of lower degree than $Q(x)$, then R(x) dx = P()Q’() |x - |, the summation applying to all the roots $\alpha$ of $Q(x) = 0$. [The form of the fraction corresponding to $\alpha$ may be deduced from the facts that Q(x)x - Q’(),0pt minus 3pt(x - ) R(x) P()Q’(), as $x \to \alpha$.]
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Exercise XLVIII, problem 4, p. 235
If all the roots of $Q(x)$ are real and $\alpha$ is a double root, the other roots being simple roots, and $P(x)$ is of lower degree than $Q(x)$, then the integral is $A/(x - \alpha) + A'\log |x - \alpha| + \sum B\log |x - \beta|$, where A = -2P()Q”(),0pt minus 3ptA’ = 23P’() Q”() - P(a) Q”’() 3Q”()^2,0pt minus 3ptB = P()Q’(), and the summation applies to all roots $\beta$ of $Q(x) = 0$ other than $\alpha$.
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Exercise XLVIII, problem 5, p. 235
Calculate dx(x - 1) (x^2 + 1)^2.
Printed answer:
- -14(x - 1) - 14(x^2 + 1) - 12 |x - 1| + 14 (x^2 + 1) + 14 x
verified: the printed answer passed a computed check
How it was checked
- integrate: passes -1/(4*(x - 1)) - 1/(4*(x**2 + 1)) - log(Abs(x - 1))/2 + log(x**2 + 1)/4 + atan(x)/4
Exercise XLVIII, problem 6a, p. 235
Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*
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- integrate: no printed answer to check
Exercise XLVIII, problem 6b, p. 235
Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*
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- (none printed)
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- integrate: no printed answer to check
Exercise XLVIII, problem 6c, p. 235
Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*
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- (none printed)
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- integrate: no printed answer to check
Exercise XLVIII, problem 6d, p. 235
Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*
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- (none printed)
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How it was checked
- integrate: no printed answer to check
Exercise XLVIII, problem 6e, p. 235
Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*
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- (none printed)
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- integrate: no printed answer to check
Exercise XLVIII, problem 6f, p. 235
Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*
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- (none printed)
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How it was checked
- integrate: no printed answer to check
Exercise XLVIII, problem 6g, p. 235
Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*
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- (none printed)
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- integrate: no printed answer to check
Exercise XLVIII, problem 6h, p. 235
Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*
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- (none printed)
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- integrate: no printed answer to check
Exercise XLVIII, problem 7a, p. 235
Prove the formulae: alignat*3 dx1 + x^4 &= 142 % &&(1 + x2 + x^21 - x2 + x^2) &&+ 2(x21 - x^2), % x^2 dx1 + x^4 &= 142 % &-&(1 + x2 + x^21 - x2 + x^2) &&+ 2(x21 - x^2), % dx1 + x^2 + x^4 &= 143% &3&(1 + x + x^21 - x + x^2) &&+ 2(x31 - x^2). alignat*
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- integrate: no printed answer to check
Exercise XLVIII, problem 7b, p. 235
Prove the formulae: alignat*3 dx1 + x^4 &= 142 % &&(1 + x2 + x^21 - x2 + x^2) &&+ 2(x21 - x^2), % x^2 dx1 + x^4 &= 142 % &-&(1 + x2 + x^21 - x2 + x^2) &&+ 2(x21 - x^2), % dx1 + x^2 + x^4 &= 143% &3&(1 + x + x^21 - x + x^2) &&+ 2(x31 - x^2). alignat*
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- (none printed)
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How it was checked
- integrate: no printed answer to check
Exercise XLVIII, problem 7c, p. 235
dx1 + x^2 + x^4
Printed answer:
- 143% &3&(1 + x + x^21 - x + x^2) &&+ 2(x31 - x^2)
verified: the printed answer passed a computed check
How it was checked
- integrate: passes 1/(4*sqrt(3))*(sqrt(3)*log((1 + x + x**2)/(1 - x + x**2)) + 2*atan(x*sqrt(3)/(1 - x**2)))

Exercise XLIX

Exercise XLIX, problem 1, p. 240
Prove that if $a > 0$ then align* x^2 + a^2 dx &= 12x x^2 + a^2 + 12a^2 x + x^2 + a^2, x^2 - a^2 dx &= 12x x^2 - a^2 - 12a^2 |x + x^2 - a^2|, a^2 - x^2 dx &= 12x a^2 - x^2 + 12a^2 (x/a). align*
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- other: not a kind the checker handles
Exercise XLIX, problem 10, p. 240
Prove that f”(x) F(x) dx = f’(x) F(x) - f(x) F’(x) + f(x) F”(x) dx and generally % multline* %[** TN: Set on one line in the original] f^(n)(x) F(x) dx = f^(n-1)(x) F(x) - f^(n-2)(x) F’(x) + … + (-1)^n f(x) F^(n)(x) dx. multline*
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Exercise XLIX, problem 11, p. 240
The integral $\ds\int (1 + x)^{p} x^{q}\, dx$, where $p$ and $q$ are rational, can be found in three cases, viz. (i) if $p$ is an integer, (ii) if $q$ is an integer, and (iii) if $p + q$ is an integer. [In case (i) put $x = u^{s}$, where $s$ is the denominator of $q$; in case (ii) put $1 + x = t^{s}$, where $s$ is the denominator of $p$; and in case (iii) put $1 + x = xt^{s}$, where $s$ is the denominator of $p$.]
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- other: not a kind the checker handles
Exercise XLIX, problem 12, p. 240
The integral $\ds\int x^{m}(ax^{n} + b)^{q}\, dx$ can be reduced to the preceding integral by the substitution $ax^{n} = bt$. [In practice it is often most convenient to calculate a particular integral of this kind by a ‘formula of reduction’ (cf. [misc:VI]Misc. Ex. 39).]
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- other: not a kind the checker handles
Exercise XLIX, problem 13, p. 240
The integral $\ds\int R\{x, \sqrtp{ax + b}, \sqrtp{cx + d}\}\, dx$ can be reduced to that of a rational function by the substitution 4x = -(b/a) t + (1/t)^2 - (d/c)t - (1/t)^2.
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- other: not a kind the checker handles
Exercise XLIX, problem 14, p. 240
Reduce $\ds\int R(x, y)\, dx$, where $y^{2}(x - y) = x^{2}$, to the integral of a rational function. [Putting $y = tx$ we obtain $x = 1/\{t^{2}(1 - t)\}$, $y = 1/\{t(1 - t)\}$.]
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- other: not a kind the checker handles
Exercise XLIX, problem 15a, p. 240
0.375em plus 0.75em minus 0.25emReduce the integral in the same way when (*a*) $y(x - y)^{2} = x$, (*b*) $(x^{2} + y^{2})^{2} = a^{2}(x^{2} - y^{2})$. [In case (*a*) put $x - y = t$: in case (b) put $x^{2} + y^{2} = t(x - y)$, when we obtain %[** TN: Set in-line in the original] x = a^2t(t^2 + a^2)/(t^4 + a^4),0pt minus 3pty = a^2t(t^2 - a^2)/(t^4 + a^4).]
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- other: not a kind the checker handles
Exercise XLIX, problem 15b, p. 240
0.375em plus 0.75em minus 0.25emReduce the integral in the same way when (*a*) $y(x - y)^{2} = x$, (*b*) $(x^{2} + y^{2})^{2} = a^{2}(x^{2} - y^{2})$. [In case (*a*) put $x - y = t$: in case (b) put $x^{2} + y^{2} = t(x - y)$, when we obtain %[** TN: Set in-line in the original] x = a^2t(t^2 + a^2)/(t^4 + a^4),0pt minus 3pty = a^2t(t^2 - a^2)/(t^4 + a^4).]
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- other: not a kind the checker handles
Exercise XLIX, problem 16, p. 240
If $y(x - y)^{2} = x$ then dxx - 3y = 12 (x - y)^2 - 1.
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- other: not a kind the checker handles
Exercise XLIX, problem 17, p. 240
If $(x^{2} + y^{2})^{2} = 2c^{2}(x^{2} - y^{2})$ then dxy(x^2 + y^2 + c^2) = - 1c^2(x^2 + y^2x - y).
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How it was checked
- other: not a kind the checker handles
Exercise XLIX, problem 2a, p. 240
Calculate the integrals $\ds\int \frac{dx}{\sqrtp{a^{2} - x^{2}}}$, $\ds\int \sqrtp{a^{2} - x^{2}}\, dx$ by means of the substitution $x = a\sin\theta$, and verify that the results agree with those obtained in [§]135 and Ex. 1.
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- (none printed)
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How it was checked
- integrate: no printed answer to check
Exercise XLIX, problem 2b, p. 240
Calculate the integrals $\ds\int \frac{dx}{\sqrtp{a^{2} - x^{2}}}$, $\ds\int \sqrtp{a^{2} - x^{2}}\, dx$ by means of the substitution $x = a\sin\theta$, and verify that the results agree with those obtained in [§]135 and Ex. 1.
Printed answer:
- (none printed)
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How it was checked
- integrate: no printed answer to check
Exercise XLIX, problem 3, p. 240
Calculate $\ds\int x(x + a)^{m}\, dx$, where $m$ is any rational number, in three ways, viz. (i) by integration by parts, (ii) by the substitution $(x + a)^{m} = t$, and (iii) by writing $(x + a) - a$ for $x$; and verify that the results agree.
Printed answer:
- (none printed)
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How it was checked
- integrate: no printed answer to check
Exercise XLIX, problem 4a, p. 240
Prove, by means of the substitutions $ax + b = 1/t$ and $x = 1/u$, that (in the notation of [§§]130 and 138) dxy^3 = ax + by,0pt minus 3ptx dxy^3 = -bx + cy.
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- other: not a kind the checker handles
Exercise XLIX, problem 4b, p. 240
Prove, by means of the substitutions $ax + b = 1/t$ and $x = 1/u$, that (in the notation of [§§]130 and 138) dxy^3 = ax + by,0pt minus 3ptx dxy^3 = -bx + cy.
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- (none printed)
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How it was checked
- other: not a kind the checker handles
Exercise XLIX, problem 5, p. 240
Calculate $\ds\int \frac{dx}{\sqrtb{(x - a) (b - x)}}$, where $b > a$, in three ways, viz. (i) by the methods of the preceding sections, (ii) by the substitution $(b - x)/(x - a) = t^{2}$, and (iii) by the substitution $x = a\cos^{2}\theta + b\sin^{2}\theta$; and verify that the results agree.
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- (none printed)
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How it was checked
- integrate: no printed answer to check
Exercise XLIX, problem 6a, p. 240
Integrate $\sqrtb{(x - a) (b - x)}$ and $\sqrtb{(b - x)/(x - a)}$.
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- integrate: no printed answer to check
Exercise XLIX, problem 6b, p. 240
Integrate $\sqrtb{(x - a) (b - x)}$ and $\sqrtb{(b - x)/(x - a)}$.
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- integrate: no printed answer to check
Exercise XLIX, problem 7, p. 240
Show, by means of the substitution $2x + a + b = \frac{1}{2}(a - b) \{t^{2} + (1/t)^{2}\}$, or by multiplying numerator and denominator by $\sqrtp{x + a} -\sqrtp{x + b}$, that if $a > b$ then dxx + a + x + b = 12a - b (t + 13t^3).
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- other: not a kind the checker handles
Exercise XLIX, problem 8, p. 240
Find a substitution which will reduce $\ds\int \frac{dx}{(x + a)^{3/2} + (x - a)^{3/2}}$ to the integral of a rational function. % [0]% (*Math. Trip.* 1899.)% [1]%
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- other: not a kind the checker handles
Exercise XLIX, problem 9, p. 240
0.375em plus 0.75em minus 0.25emShow that $\ds\int R\{x, \sqrtp[n]{ax + b}\}\, dx$ is reduced, by the substitution $ax + b = y^{n}$, to the integral of a rational function.
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- other: not a kind the checker handles

Exercise L

Exercise L, problem 10, p. 244
Show that the integral $\ds\int R(x, y)\, dx$, where $y^{2} = ax^{2} + 2bx + c$, is rationalised by the substitution $t = (x - p)/(y + q)$, where $(p, q)$ is any point on the conic $y^{2} = ax^{2} + 2bx + c$. [The integral is of course also rationalised by the substitution $t = (x - p)/(y - q)$: cf. [§]134.]
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Exercise L, problem 1a, p. 244
Evaluate dxx x^2 + 2x + 3,0pt minus 3ptdx(x - 1) x^2 + 1,0pt minus 3ptdx(x + 1) 1 + 2x - x^2.
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Exercise L, problem 1b, p. 244
Evaluate dxx x^2 + 2x + 3,0pt minus 3ptdx(x - 1) x^2 + 1,0pt minus 3ptdx(x + 1) 1 + 2x - x^2.
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Exercise L, problem 1c, p. 244
Evaluate dxx x^2 + 2x + 3,0pt minus 3ptdx(x - 1) x^2 + 1,0pt minus 3ptdx(x + 1) 1 + 2x - x^2.
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Exercise L, problem 2, p. 244
Prove that dx(x - p) (x - p) (x - q) = 2q - p x - qx - p.
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Exercise L, problem 3, p. 244
If $ag^{2} + ch^{2} = -\nu < 0$ then dx(hx + g) ax^2 + c = -1 [ (ax^2 + c)ch - agx ].
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Exercise L, problem 4, p. 244
Show that $\ds\int \frac{dx}{(x - x_{0})y}$, where $y^{2} = ax^{2} + 2bx + c$, may be expressed in one or other of the forms -1y_0 | axx_0 + b(x + x_0) + c + yy_0x - x_0 |,0pt minus 3pt1z_0 axx_0 + b(x + x_0) + cyz_0 , according as $ax_{0}^{2} + 2bx_{0} + c$ is positive and equal to $y_{0}^{2}$ or negative and equal to $-z_{0}^{2}$.
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Exercise L, problem 5, p. 244
Show by means of the substitution $y = \sqrtp{ax^{2} + 2bx + c}/(x - p)$ that dx(x - p) ax^2 + 2bx + c = dyy^2 - , where $\lambda = ap^{2} + 2bp + c$, $\mu = ac - b^{2}$. [This method of reduction is elegant but less straightforward than that explained in [§]139.]
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Exercise L, problem 6, p. 244
Show that the integral dxx 3x^2 + 2x + 1 is rationalised by the substitution $x = (1 + y^{2})/(3 - y^{2})$. % [0]% (*Math. Trip.* 1911.)% [1]%
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Exercise L, problem 7, p. 244
Calculate (x + 1) dx(x^2 + 4) x^2 + 9. [pg]245
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Exercise L, problem 8, p. 244
Calculate dx(5x^2 + 12x + 8) 5x^2 + 2x - 7.
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Exercise L, problem 9a, p. 244
Calculate (x + 1) dx(2x^2 - 2x + 1) 3x^2 - 2x + 1,0pt minus 3pt(x - 1) dx(2x^2 - 6x + 5) 7x^2 - 22x + 19. % [0]% (*Math. Trip.* 1911.)% [1]%
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Exercise L, problem 9b, p. 244
Calculate (x + 1) dx(2x^2 - 2x + 1) 3x^2 - 2x + 1,0pt minus 3pt(x - 1) dx(2x^2 - 6x + 5) 7x^2 - 22x + 19. % [0]% (*Math. Trip.* 1911.)% [1]%
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Exercise Misc-VI

Exercise Misc-VI, problem 1, p. 253
A function $f(x)$ is defined as being equal to $1 + x$ when $x \leq 0$, to $x$ when $0 < x < 1$, to $2 - x$ when $1 \leq x \leq 2$, and to $3x - x^{2}$ when $x > 2$. Discuss the continuity of $f(x)$ and the existence and continuity of $f'(x)$ for $x = 0$, $x = 1$, and $x = 2$.
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Exercise Misc-VI, problem 10, p. 253
If $ax + by + c = 0$ then $y_{2} = 0$ (suffixes denoting differentiations with respect to $x$). We may express this by saying that *the general differential equation of all straight lines is $y_{2} = 0$*. Find the general differential equations of (i) all circles with their centres on the axis of $x$, (ii) all parabolas with their axes along the axis of $x$, (iii) all parabolas with their axes parallel to the axis of $y$, (iv) all circles, (v) all parabolas, (vi) all conics.
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Exercise Misc-VI, problem 11, p. 253
Show that the general differential equations of all parabolas and of all conics are respectively D_x^2 (y_2^-2/3) = 0,0pt minus 3ptD_x^3 (y_2^-2/3) = 0.
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Exercise Misc-VI, problem 12, p. 253
Denoting $\dfrac{dy}{dx}$, $\dfrac{1}{2!}\, \dfrac{d^{2}y}{dx^{2}}$, $\dfrac{1}{3!}\, \dfrac{d^{3}y}{dx^{3}}$, $\dfrac{1}{4!}\, \dfrac{d^{4}y}{dx^{4}}$, … by $t$, $a$, $b$, $c$, … and $\dfrac{dx}{dy}$, $\dfrac{1}{2!}\, \dfrac{d^{2}x}{dy^{2}}$, $\dfrac{1}{3!}\, \dfrac{d^{3}x}{dy^{3}}$, $\dfrac{1}{4!}\, \dfrac{d^{4}x}{dy^{4}}$, … by $\tau$, $\alpha$, $\beta$, $\gamma$, …, show that 4ac - 5b^2 = (4- 5^2)/^8,0pt minus 3ptbt - a^2 = - (- ^2)/^6. Establish similar formulae for the functions $a^{2}d - 3abc - 2b^{3}$, $(1 + t^{2})b - 2a^{2}t$, $2ct - 5ab$.
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Exercise Misc-VI, problem 13, p. 253
Prove that, if $y_{k}$ is the $k$th derivative of $y = \sin(n\arcsin x)$, then (1 - x^2)y_k+2 - (2k + 1)xy_k+1 + (n^2 - k^2)y_k = 0.
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Exercise Misc-VI, problem 14, p. 253
Prove the formula vD_x^nu = D_x^n(uv) - nD_x^n-1(uD_xv) + n(n - 1)1·2 D_x^n-2(uD_x^2v) - … where $n$ is any positive integer.
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Exercise Misc-VI, problem 15, p. 253
A curve is given by x = a(2t + 2t),0pt minus 3pty = a(2t - 2t). Prove (i) that the equations of the tangent and normal, at the point $P$ whose parameter is $t$, are x12 t + y12 t = a32 t,0pt minus 3ptx12 t - y12 t = 3a32 t; (ii) that the tangent at $P$ meets the curve in the points $Q$, $R$ whose parameters are $-\frac{1}{2} t$ and $\pi - \frac{1}{2} t$; (iii) that $QR = 4a$; (iv) that the tangents at $Q$ and $R$ are at right angles and intersect on the circle $x^{2} + y^{2} = a^{2}$; (v) that the normals at $P$, $Q$, and $R$ are concurrent and intersect on the circle $x^{2} + y^{2} = 9a^{2}$; (vi) that the equation of the curve is (x^2 + y^2 + 12ax + 9a^2)^2 = 4a(2x + 3a)^3. Sketch the form of the curve.
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Exercise Misc-VI, problem 16, p. 253
Show that the equations which define the curve of Ex. 15 may be replaced by $\xi/a = 2u + (1/u^{2})$, $\eta/a = (2/u) + u^{2}$, where $\xi = x + yi$, $\eta = x - yi$, $u = \Cis t$. Show that the tangent and normal, at the point defined by $u$, are u^2- u= a(u^3 - 1),0pt minus 3ptu^2+ u= 3a(u^3 + 1), and deduce the properties (ii)--(v) of Ex. 15.
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Exercise Misc-VI, problem 17, p. 253
Show that the condition that $x^{4} + 4px^{3} - 4qx - 1 = 0$ should have equal roots may be expressed in the form $(p + q)^{2/3} - (p - q)^{2/3} = 1$.
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Exercise Misc-VI, problem 18, p. 253
The roots of a cubic $f(x) = 0$ are $\alpha$, $\beta$, $\gamma$ in ascending order of magnitude. Show that if $\DPmod{(\alpha, \beta)}{[\alpha, \beta]}$ and $\DPmod{(\beta, \gamma)}{[\beta, \gamma]}$ are each divided into six equal sub-intervals, then a root of $f'(x) = 0$ will fall in the fourth interval from $\beta$ on each side. What will be the nature of the cubic in the two cases when a root of $f'(x) = 0$ falls at a point of division?
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Exercise Misc-VI, problem 19, p. 253
Investigate the maxima and minima of $f(x)$, and the real roots of $f(x) = 0$, $f(x)$ being either of the functions x - x - (1 - x),0pt minus 3ptx - x - (- ) - 12(- x), and $\alpha$ an angle between $0$ and $\pi$. Show that in the first case the condition for a double root is that $\tan\alpha - \alpha$ should be a multiple of $\pi$.
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Exercise Misc-VI, problem 2, p. 253
Denoting $a$, $ax + b$, $ax^{2} + 2bx + c$, … by $u_{0}$, $u_{1}$, $u_{2}$, …, show that $u_{0}^{2} u_{3} - 3u_{0} u_{1} u_{2} + 2u_{1}^{3}$ and $u_{0} u_{4} - 4u_{1} u_{3} + 3u_{2}^{2}$ are independent of $x$.
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Exercise Misc-VI, problem 20, p. 253
Show that by choice of the ratio $\lambda : \mu$ we can make the roots of $\lambda(ax^{2} + bx + c) + \mu(a'x^{2} + b'x + c') = 0$ real and having a difference of any magnitude, unless the roots of the two quadratics are all real and interlace; and that in the excepted case the roots are always real, but there is a lower limit for the magnitude of their difference.
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Exercise Misc-VI, problem 21, p. 253
Prove that < xx(1 - x) 4 when $0 < x < 1$, and draw the graph of the function.
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Exercise Misc-VI, problem 22, p. 253
Draw the graph of the function x - 1x - 1x - 1.
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Exercise Misc-VI, problem 23, p. 253
Sketch the general form of the graph of $y$, given that dydx = (6x^2 + x - 1) (x - 1)^2 (x + 1)^3x^2.
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Exercise Misc-VI, problem 24, p. 253
A sheet of paper is folded over so that one corner just reaches the opposite side. Show how the paper must be folded to make the length of the crease a maximum.
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Exercise Misc-VI, problem 25, p. 253
The greatest acute angle at which the ellipse $(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1$ can be cut by a concentric circle is $\arctan\{(a^{2} - b^{2})/2ab\}$.
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Exercise Misc-VI, problem 26, p. 253
In a triangle the area $\Delta$ and the semi-perimeter $s$ are fixed. Show that any maximum or minimum of one of the sides is a root of the equation $s(x - s) x^{2} + 4\Delta^{2} = 0$. Discuss the reality of the roots of this equation, and whether they correspond to maxima or minima.
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Exercise Misc-VI, problem 27, p. 253
The area of the greatest equilateral triangle which can be drawn with its sides passing through three given points $A$, $B$, $C$ is 2+ a^2 + b^2 + c^223, $a$, $b$, $c$ being the sides and $\Delta$ the area of $ABC$.
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Exercise Misc-VI, problem 28, p. 253
If $\Delta$, $\Delta'$ are the areas of the two maximum isosceles triangles which can be described with their vertices at the origin and their base angles on the cardioid $r = a(1 + \cos\theta)$, then $256\Delta\Delta' = 25a^{4}\sqrt{5}$.
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Exercise Misc-VI, problem 29, p. 253
Find the limiting values which $(x^{2} - 4y + 8)/(y^{2} - 6x + 3)$ approaches as the point $(x, y)$ on the curve $x^{2}y - 4x^{2} - 4xy + y^{2} + 16x - 2y - 7 = 0$ approaches the position $(2, 3)$.
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Exercise Misc-VI, problem 3, p. 253
If $a_{0}$, $a_{1}$, …, $a_{2n}$ are constants and $U_{r} = (a_{0}, a_{1}, \dots, a_{r} \btw x, 1)^{r}$, then U_0U_2n - 2nU_1U_2n-1 + 2n(2n - 1)1·2 U_2U_2n-2 - …+ U_2nU_0 is independent of $x$.
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Exercise Misc-VI, problem 30, p. 253
If $f(x) = \dfrac{1}{\sin x - \sin a} - \dfrac{1}{(x - a)\cos a}$, then dda_x a f(x) - _x af’(x) = 34 ^3 a - 512 a.
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Exercise Misc-VI, problem 31, p. 253
Show that if $\phi(x) = 1/(1 + x^{2})$ then $\phi^{n} (x) = Q_{n}(x)/(1 + x^{2})^{n+1}$, where $Q_{n}(x)$ is a polynomial of degree $n$. Show also that (i) $Q_{n+1} = (1 + x^{2}) Q_{n}' - 2(n + 1) x Q_{n}$, (ii) $Q_{n+2} + 2(n + 2) x Q_{n+1} + (n + 2)(n + 1)(1 + x^{2})Q_{n} = 0$, (iii) $(1 + x^{2}) Q_{n}'' - 2nx Q_{n}' + n(n + 1)Q_{n} = 0$, (iv) $Q_{n} = (-1)^{n} n!\left\{(n + 1)x^{n} - \dfrac{(n + 1)n(n - 1)}{3!} x^{n-2} + \dots\right\}$, (v) all the roots of $Q_{n} = 0$ are real and separated by those of $Q_{n-1} = 0$.
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Exercise Misc-VI, problem 32, p. 253
If $f(x)$, $\phi(x)$, $\psi(x)$ have derivatives when $a \leq x \leq b$, then there is a value of $\xi$ lying between $a$ and $b$ and such that vmatrix f(a) & (a) & (a) f(b) & (b) & (b) f’()& ’()& ’() vmatrix =0.
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Exercise Misc-VI, problem 33, p. 253
Deduce from Ex. 32 the formula f(b) - f(a)(b) - (a) = f’()’()
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Exercise Misc-VI, problem 34, p. 253
If $\phi'(x) \to a$ as $x \to \infty$, then $\phi(x)/x \to a$. If $\phi'(x) \to \infty$ then $\phi(x) \to \infty$.
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Exercise Misc-VI, problem 35, p. 253
If $\phi(x) \to a$ as $x \to \infty$, then $\phi'(x)$ cannot tend to any limit other than zero.
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Exercise Misc-VI, problem 36, p. 253
If $\phi(x) + \phi'(x) \to a$ as $x \to \infty$, then $\phi(x) \to a$ and $\phi'(x) \to 0$.
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Exercise Misc-VI, problem 37, p. 253
Show how to reduce $\ds\int R\left\{x, \bigsqrtp{\frac{ax + b}{mx + n}}, \bigsqrtp{\frac{cx + d}{mx + n}}\right\} dx$ to the integral of a rational function.
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Exercise Misc-VI, problem 38, p. 253
dx(1 + x^2)^3,0pt minus 3ptx - 1x + 1 dxx,0pt minus 3ptx dx1 + x - [3]1 + x, a^2 + b^2 + cx dx,0pt minus 3pt^3x dx,0pt minus 3pt5x + 62x + x + 3 dx, dx(2 - ^2x) (2 + x - ^2 x),0pt minus 3ptxx dx^4x + ^4x,0pt minus 3ptx 2x dx, dx(1 + x) (2 + x),0pt minus 3ptx + x1 + x dx,0pt minus 3ptx dx,0pt minus 3pt(x)^2 dx, xx dx,0pt minus 3ptxx1 - x^2 dx,0pt minus 3ptxx^3 dx,0pt minus 3ptx(1 + x)^2 dx, xx^2 dx,0pt minus 3ptx(1 + x^2)^3/2 dx,0pt minus 3pt(^2 + ^2x^2)x^2 dx,0pt minus 3pt(+ x)(a + bx)^2 dx.
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Exercise Misc-VI, problem 39, p. 253
**of reduction.** (i) Show that $2(n - 1)(q - \tfrac{1}{4}p^{2}) \int \frac{dx}{(x^{2} + px + q)^{n}} = \frac{x + \frac{1}{2}p}{(x^{2} + px + q)^{n-1}} + (2n - 3) \int \frac{dx}{(x^{2} + px + q)^{n-1}}.$ (ii) Show that if $I_{p, q} = \ds\int x^{p}(1 + x)^{q}\, dx$ then $(p + 1) I_{p, q} = x^{p+1}(1 + x)^{q} - qI_{p+1, q-1},$ and obtain a similar formula connecting $I_{p, q}$ with $I_{p-1, q+1}$. Show also, by means of the substitution $x = -y/(1 + y)$, that $I_{p, q} = (-1)^{p+1} \int y^{p} (1 + y)^{-p-q-2}\, dy.$ (iii) Show that if $X = a + bx$ then $\int xX^{-1/3}\, dx = -3(3a - 2bx) X^{2/3}/10b^{2}$, $\int x^{2}X^{-1/3}\, dx = 3(9a^{2} - 6abx + 5b^{2}x^{2}) X^{2/3}/40b^{3}\DPchg{.}{,}$, $\int xX^{-1/4}\, dx = -4(4a - 3bx) X^{3/4}/21b^{2}$, $\int x^{2}X^{-1/4}\, dx = 4(32a^{2} - 24abx + 21b^{2}x^{2}) X^{3/4}/231b^{3}$. (iv) If $I_{m, n} = \ds\int \frac{x^{m}\, dx}{(1 + x^{2})^{n}}$ then $2(n - 1)I_{m, n} = -x^{m-1} (1 + x^{2})^{-(n-1)} + (m - 1)I_{m-2, n-1}.$ (v) If $I_{n} = \ds\int x^{n} \cos\beta x\, dx$ and $J_{n} = \ds\int x^{n} \sin\beta x\, dx$ then $\beta I_{n} = x^{n} \sin\beta x - nJ_{n-1}, \beta J_{n} = -x^{n} \cos\beta x + nI_{n-1}.$ (vi) If $I_{n} = \ds\int \cos^{n} x\, dx$ and $J_{n} = \ds\int \sin^{n} x\, dx$ then $nI_{n} = \sin x\cos^{n-1} x + (n - 1) I_{n-2}, nJ_{n} = -\cos x\sin^{n-1} x + (n - 1) J_{n-2}.$ (vii) If $I_{n} = \ds\int \tan^{n}x\, dx$ then $(n - 1)(I_{n} + I_{n-2}) = \tan^{n-1}x$. (viii) If $I_{m, n} = \ds\int \cos^{m}x \sin^{n}x\, dx$ then $(m+n)I_{m, n} = -\cos^{m+1}x \sin^{n-1}x + (n - 1) I_{m, n-2} = \cos^{m-1}x \sin^{n+1}x + (m - 1) I_{m-2, n}.$ (ix) Connect $I_{m, n} = \ds\int \sin^{m}x \sin nx\, dx$ with $I_{m-2, n}$. (x) If $I_{m, n} = \ds\int x^{m} \cosec^{n}x\, dx$ then $(n - 1)(n - 2)I_{m, n} = (n - 2)^{2}I_{m, n-2} + m(m - 1)I_{m-2, n-2} -x^{m-1} \cosec^{n-1}x \{m\sin x + (n - 2) x\cos x\}.$ (xi) If $I_{n} = \ds\int (a + b\cos x)^{-n}\, dx$ then $(n - 1)(a^{2} - b^{2}) I_{n} = -b\sin x (a + b\cos x)^{-(n-1)} + (2n - 3)aI_{n-1} - (n - 2)I_{n-2}.$ (xii) If $I_{n} = \ds\int (a\cos^{2} x + 2h\cos x\sin x + b\sin^{2}x)^{-n}\, dx$ then $4n(n + 1)(ab - h^{2})I_{n+2} - 2n(2n + 1)(a + b)I_{n+1} + 4n^{2}I_{n} = -\frac{d^{2} I_{n}}{dx^{2}}.$ (xiii) If $I_{m, n} = \ds\int x^{m}(\log x)^{n}\, dx$ then $(m + 1)I_{m, n} = x^{m+1}(\log x)^{n} - nI_{m, n-1}.$
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Exercise Misc-VI, problem 4, p. 253
The first three derivatives of the function $\arcsin(\mu\sin x) - x$, where $\mu > 1$, are positive when $0 \leq x \leq \frac{1}{2} \pi$.
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Exercise Misc-VI, problem 40, p. 253
If $n$ is a positive integer then the value of $\ds\int x^{m}(\log x)^{n}\, dx$ is x^m+1 (x)^nm + 1 - n(x)^n-1(m + 1)^2 + n(n - 1)(x)^n-2(m + 1)^3 - …+ (-1)^nn!(m + 1)^n+1.
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Exercise Misc-VI, problem 41, p. 253
Show that the most general function $\phi(x)$, such that $\phi'' + a^{2}\phi = 0$ for all values of $x$, may be expressed in either of the forms $A\cos ax + B\sin ax$, $\rho\cos(ax + \epsilon)$, where $A$, $B$, $\rho$, $\epsilon$ are constants.
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Exercise Misc-VI, problem 42, p. 253
Determine the most general functions $y$ and $z$ such that $y' + \omega z = 0$, and $z' - \omega y = 0$, where $\omega$ is a constant and dashes denote differentiation with respect to $x$.
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Exercise Misc-VI, problem 43, p. 253
The area of the curve given by x = + 1 - ^2^2,0pt minus 3pty = - 1 - ^2^2, where $\alpha$ is a positive acute angle, is $\frac{1}{2}\pi(1 + \sin\alpha)^{2}/\sin\alpha$.
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Exercise Misc-VI, problem 44, p. 253
The projection of a chord of a circle of radius $a$ on a diameter is of constant length $2a\cos\beta$; show that the locus of the middle point of the chord consists of two loops, and that the area of either is $a^{2}(\beta - \cos\beta\sin\beta)$.
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Exercise Misc-VI, problem 45, p. 253
Show that the length of a quadrant of the curve $(x/a)^{2/3} + (y/b)^{2/3} = 1$ is $(a^{2} + ab + b^{2})/(a + b)$.
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Exercise Misc-VI, problem 46, p. 253
A point $A$ is inside a circle of radius $a$, at a distance $b$ from the centre. Show that the locus of the foot of the perpendicular drawn from $A$ to a tangent to the circle encloses an area $\pi(a^{2} + \frac{1}{2}b^{2})$.
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Exercise Misc-VI, problem 47, p. 253
Prove that if $(a, b, c, f, g, h \btw x, y, 1)^{2} = 0$ is the equation of a conic, then dx(lx + my + n)(hx + by + f) = PTPT’ + , where $PT$, $PT'$ are the perpendiculars from a point $P$ of the conic on the tangents at the ends of the chord $lx + my + n = 0$, and $\alpha$, $\beta$ are constants.
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Exercise Misc-VI, problem 48, p. 253
Show that ax^2 + 2bx + c(Ax^2 + 2Bx + C)^2 dx will be a rational function of $x$ if and only if one or other of $AC - B^{2}$ and $aC + cA - 2bB$ is zero.
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Exercise Misc-VI, problem 49, p. 253
Show that the necessary and sufficient condition that f(x)F(x)^2 dx, where $f$ and $F$ are polynomials of which the latter has no repeated factor, should be a rational function of $x$, is that $f'F' - fF''$ should be divisible by $F$.
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Exercise Misc-VI, problem 5, p. 253
The constituents of a determinant are functions of $x$. Show that its differential coefficient is the sum of the determinants formed by differentiating the constituents of one row only, leaving the rest unaltered.
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Exercise Misc-VI, problem 50, p. 253
Show that x + x + (1 - ex)^2 dx is a rational function of $\cos x$ and $\sin x$ if and only if $\alpha e + \gamma = 0$; and determine the integral when this condition is satisfied.
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Exercise Misc-VI, problem 6, p. 253
If $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$ are polynomials of degree not greater than $4$, then vmatrix f_1& f_2& f_3& f_4 f_1’& f_2’& f_3’& f_4’ f_1”& f_2”& f_3”& f_4” f_1”’& f_2”’& f_3”’& f_4”’ vmatrix is also a polynomial of degree not greater than $4$.
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Exercise Misc-VI, problem 7, p. 253
If $y^{3} + 3yx + 2x^{3} = 0$ then $x^{2}(1 + x^{3})y'' - \frac{3}{2}xy' + y = 0$.
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Exercise Misc-VI, problem 8, p. 253
Verify that the differential equation $y = \phi\{\psi(y_{1})\} + \phi\{x - \psi(y_{1})\}$, where $y_{1}$ is the derivative of $y$, and $\psi$ is the function inverse to $\phi'$, is satisfied by $y = \phi(c) + \phi(x - c)$ or by $y = 2\phi(\frac{1}{2}x)$.
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Exercise Misc-VI, problem 9, p. 253
Verify that the differential equation $y = \{x/\psi(y_{1})\} \phi\{\psi(y_{1})\}$, where the notation is the same as that of Ex. 8, is satisfied by $y = c\phi(x/c)$ or by $y = \beta x$, where $\beta = \phi(\alpha)/\alpha$ and $\alpha$ is any root of the equation $\phi(\alpha) - \alpha\phi'(\alpha) = 0$.
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Exercise LI

Exercise LI, problem 1, p. 246
Integrate $\sin^{3} x \cos^{2} 2x$.
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- - 716 x + 5483x - 3805x + 11127x.
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- integrate: passes -7*cos(x)/16 + 5*cos(3*x)/48 - 3*cos(5*x)/80 + cos(7*x)/112
Exercise LI, problem 2a, p. 246
Integrate by any method $\cos ax \cos bx$, $\sin ax \sin bx$, $\cos ax \sin bx$, $\cos^{2}x$, $\sin^{3}x$, $\cos^{4}x$, $\cos x \cos 2x \cos 3x$, $\cos^{3}2x \sin^{2}3x$, $\cos^{5}x \sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]
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Exercise LI, problem 2b, p. 246
Integrate by any method $\cos ax \cos bx$, $\sin ax \sin bx$, $\cos ax \sin bx$, $\cos^{2}x$, $\sin^{3}x$, $\cos^{4}x$, $\cos x \cos 2x \cos 3x$, $\cos^{3}2x \sin^{2}3x$, $\cos^{5}x \sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]
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Exercise LI, problem 2c, p. 246
Integrate by any method $\cos ax \cos bx$, $\sin ax \sin bx$, $\cos ax \sin bx$, $\cos^{2}x$, $\sin^{3}x$, $\cos^{4}x$, $\cos x \cos 2x \cos 3x$, $\cos^{3}2x \sin^{2}3x$, $\cos^{5}x \sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]
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Exercise LI, problem 2d, p. 246
Integrate by any method $\cos ax \cos bx$, $\sin ax \sin bx$, $\cos ax \sin bx$, $\cos^{2}x$, $\sin^{3}x$, $\cos^{4}x$, $\cos x \cos 2x \cos 3x$, $\cos^{3}2x \sin^{2}3x$, $\cos^{5}x \sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]
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Exercise LI, problem 2e, p. 246
Integrate by any method $\cos ax \cos bx$, $\sin ax \sin bx$, $\cos ax \sin bx$, $\cos^{2}x$, $\sin^{3}x$, $\cos^{4}x$, $\cos x \cos 2x \cos 3x$, $\cos^{3}2x \sin^{2}3x$, $\cos^{5}x \sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]
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Exercise LI, problem 2f, p. 246
Integrate by any method $\cos ax \cos bx$, $\sin ax \sin bx$, $\cos ax \sin bx$, $\cos^{2}x$, $\sin^{3}x$, $\cos^{4}x$, $\cos x \cos 2x \cos 3x$, $\cos^{3}2x \sin^{2}3x$, $\cos^{5}x \sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]
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Exercise LI, problem 2g, p. 246
Integrate by any method $\cos ax \cos bx$, $\sin ax \sin bx$, $\cos ax \sin bx$, $\cos^{2}x$, $\sin^{3}x$, $\cos^{4}x$, $\cos x \cos 2x \cos 3x$, $\cos^{3}2x \sin^{2}3x$, $\cos^{5}x \sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]
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Exercise LI, problem 2h, p. 246
Integrate by any method $\cos ax \cos bx$, $\sin ax \sin bx$, $\cos ax \sin bx$, $\cos^{2}x$, $\sin^{3}x$, $\cos^{4}x$, $\cos x \cos 2x \cos 3x$, $\cos^{3}2x \sin^{2}3x$, $\cos^{5}x \sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]
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Exercise LI, problem 2i, p. 246
Integrate by any method $\cos ax \cos bx$, $\sin ax \sin bx$, $\cos ax \sin bx$, $\cos^{2}x$, $\sin^{3}x$, $\cos^{4}x$, $\cos x \cos 2x \cos 3x$, $\cos^{3}2x \sin^{2}3x$, $\cos^{5}x \sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]
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Exercise XLV

Exercise XLV, problem 1, p. 215
If $\phi(x) = x^{m}$ then ^(n)(x) = m(m - 1) …(m - n + 1)x^m-n. This result enables us to write down the $n$th derivative of any polynomial.
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Exercise XLV, problem 10, p. 215
If $U_{n}$ denotes the $n$th derivative of $(Lx + M)/(x^{2} - 2Bx + C)$, then x^2 - 2Bx + C(n + 1)(n + 2) U_n+2 + 2(x - B)n + 1 U_n+1 + U_n = 0. % [0]% (*Math. Trip.* 1900.)% [1]% [First obtain the equation when $n = 0$; then differentiate $n$ times by Leibnitz’Leibniz’ Theorem.]
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Exercise XLV, problem 11, p. 215
**$n$th derivatives of $a/(a^{2} + x^{2})$ and $x/(a^{2} + x^{2})$.** Since aa^2 + x^2 = 12i (1x - ai - 1x + ai), 0pt minus 3ptxa^2 + x^2 = 12 (1x - ai + 1x + ai), we have D_x^n (aa^2 + x^2) = (-1)^n n!2i 1(x - ai)^n+1 - 1(x + ai)^n+1 , 0.375em plus 0.75em minus 0.25emand a similar formula for $D_{x}^{n}\{x/(a^{2} + x^{2})\}$. If $\rho = \sqrtp{x^{2} + a^{2}}$, and $\theta$ is the numerically smallest angle whose cosine and sine are $x/\rho$ and $a/\rho$, then $x + ai = \rho\Cis\theta$ and $x - ai = \rho\Cis(-\theta )$, and so align* D_x^n a/(a^2 + x^2) &= (-1)^n n!/2i ^-n-1 [(n + 1) - -(n + 1)] &= (-1)^n n! (x^2 + a^2)^-(n+1)/2 (n + 1) (a/x). align* Similarly D_x^n x/(a^2 + x^2) = (-1)^n n! (x^2 + a^2)^-(n+1)/2 (n + 1) (a/x).
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Exercise XLV, problem 12, p. 215
Prove that align* D_x^n (x)/x &= P_n (x + 12n) + Q_n (x + 12n)/x^n+1, D_x^n (x)/x &= P_n (x + 12n) - Q_n (x + 12n)/x^n+1, align* where $P_{n}$ and $Q_{n}$ are polynomials in $x$ of degree $n$ and $n-1$ respectively.
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Exercise XLV, problem 13, p. 215
Establish the formulae gather* %[** TN: Set on one line in the orignal] dxdy = 1 /(dydx),0pt minus 3ptd^2 xdy^2 = -d^2 ydx^2 / (dydx)^3, d^3 xdy^3 = -d^3 ydx^3 dydx - 3(d^2 ydx^2) / (dydx)^5. gather*
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Exercise XLV, problem 14, p. 215
If $yz = 1$ and $y_{r} = (1/r!) D_{x}^{r}y$, $z_{s} = (1/s!) D_{x}^{s}z$, then 1z^3 vmatrix z & z_1& z_2 z_1& z_2& z_3 z_2& z_3& z_4 vmatrix = 1y^2 vmatrix y_2& y_3 y_3& y_4 vmatrix. % [0]% (*Math. Trip.* 1905.)% [1]%
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Exercise XLV, problem 15, p. 215
If W(y, z, u) = vmatrix y & z & u y’ & z’ & u’ y”& z”& u” vmatrix, dashes denoting differentiations with respect to $x$, then W(y, z, u) = y^3 W(1, zy, uy).
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Exercise XLV, problem 16, p. 215
If ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0, then dy/dx = -(ax + hy + g)/(hx + by + f) and d^2y/dx^2 = (abc + 2fgh - af^2 - bg^2 - ch^2)/(hx + by + f)^3.
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Exercise XLV, problem 2, p. 215
If $\phi(x) = (ax + b)^{m}$ then ^(n)(x) = m(m - 1) …(m - n + 1)a^n(ax + b)^m-n. In these two examples $m$ may have any rational value. If $m$ is a positive integer, and $n > m$, then $\phi^{(n)}(x) = 0$.
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Exercise XLV, problem 3, p. 215
The formula (ddx)^n A(x - )^p = (-1)^n p(p + 1) …(p + n - 1)A(x - )^p+n enables us to write down the $n$th derivative of any rational function expressed in the standard form as a sum of partial fractions.
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Exercise XLV, problem 4, p. 215
Prove that the $n$th derivative of $1/(1 - x^{2})$ is 12(n!) (1 - x)^-n-1 + (-1)^n(1 + x)^-n-1.
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Exercise XLV, problem 5, p. 215
**’ Theorem.** If $y$ is a product $uv$, and we can form the first $n$ derivatives of $u$ and $v$, then we can form the $n$th derivative of $y$ by means of *Leibniz’ Theorem*, which gives the rule (uv)_n = u_nv + n1u_n-1v_1 + n2u_n-2v_2 + … + nru_n-rv_r + …+ uv_n, where suffixes indicate differentiations, so that $u_{n}$, for example, denotes the $n$th derivative of $u$. To prove the theorem we observe that align* (uv)_1 &= u_1v + uv_1, (uv)_2 &= u_2v + 2u_1v_1 + uv_2, align* and so on. It is obvious that by repeating this process we arrive at a formula of the type (uv)_n = u_nv + a_n, 1 u_n-1 v_1 + a_n, 2 u_n-2 v_2 + … + a_n, r u_n-r v_r + …+ uv_n. Let us assume that $a_{n, r} = \dbinom{n}{r}$ for $r = 1$, $2$, … $n - 1$, and show that if this is so then $a_{n+1, r} = \dbinom{n + 1}{r}$ for $r = 1$, $2$, … $n$. It will then follow by the principle of mathematical induction that $a_{n, r} = \dbinom{n}{r}$ for all values of $n$ and $r$ in question. When we form $(uv)_{n+1}$ by differentiating $(uv)_{n}$ it is clear that the coefficient of $u_{n+1-r}v_{r}$ is a_n, r + a_n, r-1 = nr + nr - 1 = n + 1r. This establishes the theorem.
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Exercise XLV, problem 6, p. 215
The $n$th derivative of $x^{m}f(x)$ is multline* m!(m - n)! x^m-n f(x) + n m!(m - n + 1)! x^m-n+1 f’(x) + n(n - 1)1·2 m!(m - n + 2)! x^m-n+2 f”(x) + …, multline* the series being continued for $n + 1$ terms or until it terminates.
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Exercise XLV, problem 7, p. 215
Prove that $D_{x}^{n}\cos x = \cos(x + \frac{1}{2}n\pi)$, $D_{x}^{n}\sin x = \sin(x + \frac{1}{2}n\pi)$
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Exercise XLV, problem 8, p. 215
If $y = A\cos mx + B\sin mx$ then $D_{x}^{2} y + m^{2} y = 0$. And if y = Amx + Bmx + P_n(x), where $P_{n}(x)$ is a polynomial of degree $n$, then $D_{x}^{n+3} y + m^{2} D_{x}^{n+1} y = 0$.
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Exercise XLV, problem 9, p. 215
If $x^{2} D_{x}^{2}y + x D_{x} y + y = 0$ then x^2 D_x^n+2 y + (2n + 1)x D_x^n+1 y + (n^2 + 1) D_x^n y = 0. [Differentiate $n$ times by Leibnitz’Leibniz’ Theorem.]
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Exercise XLVI

Exercise XLVI, problem 10, p. 222
Discuss similarly the function $(x - a) (x - b)^{2} (x - c)^{3}$, distinguishing the different forms of the graph which correspond to different hypotheses as to the relative magnitudes of $a$, $b$, $c$.
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Exercise XLVI, problem 11, p. 222
Show that $(ax + b)/(cx + d)$ has no maxima or minima, whatever values $a$, $b$, $c$, $d$ may have. Draw a graph of the function.
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Exercise XLVI, problem 12, p. 222
Discuss the maxima and minima of the function y = (ax^2 + 2bx + c)/(Ax^2 + 2Bx + c), when the denominator has complex roots.
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Exercise XLVI, problem 13, p. 222
The maximum and minimum values themselves are the values of $\lambda$ for which $ax^{2} + 2bx + c - \lambda(Ax^{2} + 2Bx + C)$ is a perfect square.
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Exercise XLVI, problem 14, p. 222
In general the maxima and maxima of $R(x) = P(x)/Q(x)$ are among the values of $\lambda$ obtained by expressing the condition that $P(x) - \lambda Q(x) = 0$ should have a pair of equal roots.
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Exercise XLVI, problem 15, p. 222
If $Ax^{2} + 2Bx + C = 0$ has real roots then it is convenient to proceed as follows. We have y - (a/A) = (x + )/A(Ax^2 + 2Bx + C), where $\lambda = bA - aB$, $\mu = cA - aC$.
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Exercise XLVI, problem 16, p. 222
Show that $(x - \alpha)(x - \beta)/(x - \gamma)$ assumes all real values as $x$ varies, if $\gamma$ lies between $\alpha$ and $\beta$, and otherwise assumes all values except those included in an interval of length $4\sqrtp{|\alpha - \gamma||\beta - \gamma|}$.
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Exercise XLVI, problem 17, p. 222
Show that y = x^2 + 2x + cx^2 + 4x + 3c can assume any real value if $0 < c < 1$, and draw a graph of the function in this case. % [0]% (*Math. Trip.* 1910.)% [1]%
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Exercise XLVI, problem 18, p. 222
Determine the function of the form $(ax^{2} + 2bx + c)/(Ax^{2} + 2Bx + C)$ which has turning values (*i.e.* maxima or minima) $2$ and $3$ when $x = 1$ and $x = -1$ respectively, and has the value $2.5$ when $x = 0$. % [0]% (*Math. Trip.* 1908.)% [1]%
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Exercise XLVI, problem 19, p. 222
The maximum and minimum of $(x + a) (x + b)/(x - a) (x - b)$, where $a$ and $b$ are positive, are -(a + ba - b)^2,0pt minus 3pt-(a - ba + b)^2.
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Exercise XLVI, problem 1a, p. 222
Verify Theorem B when $\phi(x) = (x - a)^{m} (x - b)^{n}$ or $\phi(x) = (x - a)^{m} (x - b)^{n} (x - c)^{p}$, where $m$, $n$, $p$ are positive integers and $a < b < c$.
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Exercise XLVI, problem 1b, p. 222
Verify Theorem B when $\phi(x) = (x - a)^{m} (x - b)^{n}$ or $\phi(x) = (x - a)^{m} (x - b)^{n} (x - c)^{p}$, where $m$, $n$, $p$ are positive integers and $a < b < c$.
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Exercise XLVI, problem 2, p. 222
Show that the polynomials 2x^3 + 3x^2 - 12x + 7,0pt minus 3pt3x^4 + 8x^3 - 6x^2 - 24x + 19 are positive when $x > 1$.
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Exercise XLVI, problem 20, p. 222
The maximum value of $(x - 1)^{2}/(x + 1)^{3}$ is $\frac{2}{27}$.
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Exercise XLVI, problem 21a, p. 222
Discuss the maxima and minima of gather* x(x - 1)/(x^2 + 3x + 3),0pt minus 3ptx^4/(x - 1)(x - 3)^3, (x - 1)^2(3x^2 - 2x - 37)/(x + 5)^2(3x^2 - 14x - 1). gather*
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Exercise XLVI, problem 21b, p. 222
Discuss the maxima and minima of gather* x(x - 1)/(x^2 + 3x + 3),0pt minus 3ptx^4/(x - 1)(x - 3)^3, (x - 1)^2(3x^2 - 2x - 37)/(x + 5)^2(3x^2 - 14x - 1). gather*
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Exercise XLVI, problem 21c, p. 222
Discuss the maxima and minima of gather* x(x - 1)/(x^2 + 3x + 3),0pt minus 3ptx^4/(x - 1)(x - 3)^3, (x - 1)^2(3x^2 - 2x - 37)/(x + 5)^2(3x^2 - 14x - 1). gather*
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Exercise XLVI, problem 22, p. 222
Find the maxima and minima of $a\cos x + b\sin x$. Verify the result by expressing the function in the form $A\cos(x - a)$.
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Exercise XLVI, problem 23a, p. 222
Find the maxima and minima of a^2^2 x + b^2^2 x,0pt minus 3ptA^2x + 2Hxx + B^2 x.
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Exercise XLVI, problem 23b, p. 222
Find the maxima and minima of a^2^2 x + b^2^2 x,0pt minus 3ptA^2x + 2Hxx + B^2 x.
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Exercise XLVI, problem 24, p. 222
Show that $\sin(x + a)/\sin(x + b)$ has no maxima or minima. Draw a graph of the function.
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Exercise XLVI, problem 25, p. 222
Show that the function ^2x(x + a)(x + b)0pt minus 3pt(0 < a < b < ) has an infinity of minima equal to $0$ and of maxima equal to -4ab/^2(a - b). % [0]% (*Math. Trip.* 1909.)% [1]%
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Exercise XLVI, problem 26, p. 222
The least value of $a^{2}\sec^{2}x + b^{2}\cosec^{2}x$ is $(a + b)^{2}$.
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Exercise XLVI, problem 27, p. 222
Show that $\tan 3x \cot 2x$ cannot lie between $\frac{1}{9}$ and $\frac{3}{2}$.
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Exercise XLVI, problem 28, p. 222
Show that, if the sum of the lengths of the hypothenuse and another side of a right-angled triangle is given, then the area of the triangle is a maximum when the angle between those sides is $60°$. % [0]% (*Math. Trip.* 1909.)% [1]%
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Exercise XLVI, problem 29a, p. 222
A line is drawn through a fixed point $(a, b)$ to meet the axes $OX$, $OY$ in $P$ and $Q$. Show that the minimum values of $PQ$, $OP + OQ$, and $OP·OQ$ are respectively $(a^{2/3} + b^{2/3})^{3/2}$, $(\sqrt{a} + \sqrt{b})^{2}$, and $4ab$.
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Exercise XLVI, problem 29b, p. 222
A line is drawn through a fixed point $(a, b)$ to meet the axes $OX$, $OY$ in $P$ and $Q$. Show that the minimum values of $PQ$, $OP + OQ$, and $OP·OQ$ are respectively $(a^{2/3} + b^{2/3})^{3/2}$, $(\sqrt{a} + \sqrt{b})^{2}$, and $4ab$.
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Exercise XLVI, problem 29c, p. 222
A line is drawn through a fixed point $(a, b)$ to meet the axes $OX$, $OY$ in $P$ and $Q$. Show that the minimum values of $PQ$, $OP + OQ$, and $OP·OQ$ are respectively $(a^{2/3} + b^{2/3})^{3/2}$, $(\sqrt{a} + \sqrt{b})^{2}$, and $4ab$.
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Exercise XLVI, problem 30, p. 222
A tangent to an ellipse meets the axes in $P$ and $Q$. Show that the least value of $PQ$ is equal to the sum of the semiaxes of the ellipse.
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Exercise XLVI, problem 31, p. 222
Find the lengths and directions of the axes of the conic ax^2 + 2hxy + by^2 = 1.
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Exercise XLVI, problem 32, p. 222
The greatest value of $x^{m}y^{n}$, where $x$ and $y$ are positive and $x + y = k$, is m^m n^n k^m+n/(m + n)^m+n.
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Exercise XLVI, problem 33, p. 222
The greatest value of $ax + by$, where $x$ and $y$ are positive and $x^{2} + xy + y^{2} = 3\kappa^{2}$, is 2a^2 - ab + b^2.
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Exercise XLVI, problem 34, p. 222
If $\theta$ and $\phi$ are acute angles connected by the relation $a \sec\theta + b \sec\phi = c$, where $a$, $b$, $c$ are positive, then $a\cos\theta + b\cos\phi$ is a minimum when $\theta = \phi$.
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Exercise XLVI, problem 3a, p. 222
Show that $x - \sin x$ is an increasing function throughout any interval of values of $x$, and that $\tan x - x$ increases as $x$ increases from $-\frac{1}{2}\pi$ to $\frac{1}{2}\pi$. For what values of $a$ is $ax - \sin x$ a steadily increasing or decreasing function of $x$?
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Exercise XLVI, problem 3b, p. 222
Show that $x - \sin x$ is an increasing function throughout any interval of values of $x$, and that $\tan x - x$ increases as $x$ increases from $-\frac{1}{2}\pi$ to $\frac{1}{2}\pi$. For what values of $a$ is $ax - \sin x$ a steadily increasing or decreasing function of $x$?
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Exercise XLVI, problem 3c, p. 222
Show that $x - \sin x$ is an increasing function throughout any interval of values of $x$, and that $\tan x - x$ increases as $x$ increases from $-\frac{1}{2}\pi$ to $\frac{1}{2}\pi$. For what values of $a$ is $ax - \sin x$ a steadily increasing or decreasing function of $x$?
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Exercise XLVI, problem 4, p. 222
Show that $\tan x - x$ also increases from $x = \frac{1}{2}\pi$ to $x = \frac{3}{2}\pi$, from $x = \frac{3}{2}\pi$ to $x = \frac{5}{2}\pi$, and so on, and deduce that there is one and only one root of the equation $\tan x = x$ in each of these intervals (cf. % [examples:xvii]Ex. xvii%. 4).
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Exercise XLVI, problem 5, p. 222
0.375em plus 0.75em minus 0.25emDeduce from Ex. 3 that $\sin x - x < 0$ if $x > 0$, from this that $\cos x - 1 + \frac{1}{2}x^{2} > 0$, and from this that $\sin x - x + \frac{1}{6} x^{3} > 0$. And, generally, prove that if align* C_2m & = x - 1 + x^22! - …- (-1)^m x^2m2m!(2m)!, S_2m+1& = x - x + x^33! - …- (-1)^m x^2m+1(2m+1)!, align* and $x> 0$, then $C_{2m}$ and $S_{2m+1}$ are positive or negative according as $m$ is odd or even.
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Exercise XLVI, problem 6, p. 222
If $f(x)$ and $f''(x)$ are continuous and have the same sign at every point of an interval $\DPmod{(a, b)}{[a, b]}$, then this interval can include at most one root of either of the equations $f(x) = 0$, $f'(x) = 0$.
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Exercise XLVI, problem 7, p. 222
The functions $u$, $v$ and their derivatives $u'$, $v'$ are continuous throughout a certain interval of values of $x$, and $uv' - u'v$ never vanishes at any point of the interval. Show that between any two roots of $u = 0$ lies one of $v = 0$, and conversely. Verify the theorem when $u = \cos x$, $v = \sin x$.
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Exercise XLVI, problem 8a, p. 222
Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.
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Exercise XLVI, problem 8b, p. 222
Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.
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Exercise XLVI, problem 8c, p. 222
Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.
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Exercise XLVI, problem 8d, p. 222
Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.
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Exercise XLVI, problem 8e, p. 222
Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.
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Exercise XLVI, problem 8f, p. 222
Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.
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Exercise XLVI, problem 9, p. 222
Discuss the maxima and minima of the function $(x - a)^{m} (x - b)^{n}$, where $m$ and $n$ are any positive integers, considering the different cases which occur according as $m$ and $n$ are odd or even. Sketch the graph of the function.
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