Dodges, Pitfalls, and Triumphs

Excerpts

Dodges, Pitfalls, and Triumphs
Write $u = w$, and for $\sin w · dw$ write $dx$. We shall then have $du = dw$, while $\ds\int \sin w · dw = -\cos w = x$.
Dodges, Pitfalls, and Triumphs
There are whole treatises, such as Boole’s *Differential Equations*, devoted to the subject of thus finding the “solutions” for different original forms.
Dodges, Pitfalls, and Triumphs
A great part of the labour of integrating things consists in licking them into some shape that can be integrated.
Dodges, Pitfalls, and Triumphs
It is useful in some cases that you can’t tackle directly, for it shows that if in any case $\ds\int x\, du$ can be found, then $\ds\int u\, dx$ can also be found.
Dodges, Pitfalls, and Triumphs
A beginner is liable to overlook certain points that a practised hand would avoid; such as the use of factors that are equivalent to either zero or infinity, and the occurrence of indeterminate quantities such as $\tfrac{0}{0}$. There is no golden rule that will meet every possible case. Nothing but practice and intelligent care will avail.
Dodges, Pitfalls, and Triumphs
The solution often seems as different from the original expression as a butterfly does from the caterpillar that it was.
Dodges, Pitfalls, and Triumphs
Generally it is much easier to state the appropriate differential equation than to solve it:---the real trouble begins then only when one wants to integrate, unless indeed the equation is seen to possess some standard form of which the integral is known, and then the triumph is easy.
Dodges, Pitfalls, and Triumphs
By triumphs must be understood the successes with which the calculus has been applied to the solution of problems otherwise intractable.
Dodges, Pitfalls, and Triumphs
Notice that the same integral can be expressed sometimes in more than one way (which are equivalent to one another).

Equations

Dodges, Pitfalls, and Triumphs
\int u\, dx = ux - \int x\, du + C.
The integral of u dx equals ux minus the integral of x du, plus a constant, so an integral that is hard to find directly can be traded for one that may be easier. In this book x denotes the antiderivative of the second factor, as in the chapter's examples.
Dodges, Pitfalls, and Triumphs
d(ux) = u\, dx + x\, du,
The differential of the product ux equals u dx plus x du, which is the product rule written in differentials and is the basis of integration by parts.
Dodges, Pitfalls, and Triumphs
\int \sqrt{1-x^2}\, dx = \frac{x \sqrt{1-x^2}}{2} + \tfrac{1}{2} \arcsin x +C.
The integral of the square root of 1 minus x squared equals half of x times that square root plus half of arcsin x, plus a constant, obtained from the chapter's integration-by-parts and dodge argument.
Dodges, Pitfalls, and Triumphs
\frac{1}{a^2-x^2} = \frac{1}{2a(a+x)} + \frac{1}{2a(a-x)},
The function 1 over (a squared minus x squared) splits into the sum of two simpler fractions, 1 over 2a(a+x) and 1 over 2a(a-x).
Dodges, Pitfalls, and Triumphs
\dfrac{dy}{dx} = \dfrac{1}{a^2-x^2}
The derivative of y with respect to x equals 1 over (a squared minus x squared); this is a differential equation whose solution is found by integration.
Dodges, Pitfalls, and Triumphs
y = \dfrac{1}{2a} \log_\epsilon \dfrac{a+x}{a-x} + C?
The solution of the differential equation dy/dx = 1/(a squared minus x squared) is y equal to 1 over 2a times the natural logarithm of (a+x)/(a-x), plus a constant.
Dodges, Pitfalls, and Triumphs
u=\dfrac{1}{a} \arctan \dfrac{u}{a}
As printed, this says u equals (1/a) arctan(u/a); the book means that differentiating (1/a) arctan(u/a) gives du/(u squared + a squared), so the equation is the integral result and is not a literal identity in u. Flagged for the book's loose notation.

Problems

Exercise XIX

Exercise XIX, problem 1, p. 233
Find $\ds\int \sqrt {a^2 - x^2}\, dx$.
Printed answer:
- $\dfrac{x\sqrt{a^2 - x^2}}{2} + \dfrac{a^2}{2} \sin^{-1} \dfrac{x}{a} + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: passes x*sqrt(a**2 - x**2)/2 + a**2/2*asin(x/a)
Exercise XIX, problem 10, p. 233
Find $\ds\int \dfrac{(x^2 -3)\, dx}{x^3 - 7x+6}$.
Printed answer:
- $\frac{1}{2} \log_\epsilon(x - 1) + \frac{1}{5} \log_\epsilon(x - 2) + \frac{3}{10} \log_\epsilon(x + 3) + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: PASS-ABS-CONVENTION Rational(1, 2)*log(x - 1) + Rational(1, 5)*log(x - 2) + Rational(3, 10)*log(x + 3)
Exercise XIX, problem 11, p. 233
Find $\ds\int \dfrac{b\, dx}{x^2 -a^2}$.
Printed answer:
- $\dfrac{b}{2a} \log_\epsilon \dfrac{x - a}{x + a} + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: PASS-ABS-CONVENTION b/(2*a)*log((x - a)/(x + a))
Exercise XIX, problem 12, p. 233
Find $\ds\int \dfrac{4x\, dx}{x^4 -1}$.
Printed answer:
- $\log_\epsilon \dfrac{x^2 - 1}{x^2 + 1} + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: passes log((x**2 - 1)/(x**2 + 1))
Exercise XIX, problem 13, p. 233
Find $\ds\int \dfrac{dx}{1-x^4}$.
Printed answer:
- $\frac{1}{4} \log_\epsilon \dfrac{1 + x}{1 - x} + \frac{1}{2} \arctan x + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: PASS-ABS-CONVENTION Rational(1, 4)*log((1 + x)/(1 - x)) + Rational(1, 2)*atan(x)
Exercise XIX, problem 14, p. 233
Find $\ds\int \dfrac{dx}{x \sqrt {a-bx^2}}$.
Printed answer:
- $\dfrac{1}{\sqrt{a}} \log_\epsilon \dfrac{\sqrt{a} - \sqrt{a - bx^2}}{x\sqrt{a}}$
verified: the printed answer passed a computed check
How it was checked
- integrate: passes 1/sqrt(a)*log((sqrt(a) - sqrt(a - b*x**2))/(x*sqrt(a)))
Exercise XIX, problem 2, p. 233
Find $\ds\int x \log_\epsilon x\, dx$.
Printed answer:
- $\dfrac{x^2}{2}(\log_\epsilon x - \tfrac{1}{2}) + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: passes x**2/2*(log(x) - Rational(1, 2))
Exercise XIX, problem 3, p. 233
Find $\ds\int x^a \log_\epsilon x\, dx$.
Printed answer:
- $\dfrac{x^{a+1}}{a + 1} \left(\log_\epsilon x - \dfrac{1}{a + 1}\right) + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: passes x**(a + 1)/(a + 1)*(log(x) - 1/(a + 1))
Exercise XIX, problem 4, p. 233
Find $\ds\int \epsilon^x \cos \epsilon^x\, dx$.
Printed answer:
- $\sin \DPtypo{e}{\epsilon}^x + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: passes sin(exp(x))
Exercise XIX, problem 5, p. 233
Find $\ds\int \dfrac{1}{x} \cos (\log_\epsilon x)\, dx$.
Printed answer:
- $\sin(\log_\epsilon x) + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: passes sin(log(x))
Exercise XIX, problem 6, p. 233
Find $\ds\int x^2 \epsilon^x\, dx$.
Printed answer:
- $\DPtypo{e}{\epsilon}^x (x^2 - 2x + 2) + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: passes exp(x)*(x**2 - 2*x + 2)
Exercise XIX, problem 7, p. 233
Find $\ds\int \dfrac{(\log_\epsilon x)^a}{x}\, dx$.
Printed answer:
- $\dfrac{1}{a + 1} (\log_\epsilon x)^{a+1} + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: passes log(x)**(a + 1)/(a + 1)
Exercise XIX, problem 8, p. 233
Find $\ds\int \dfrac{dx}{x \log_\epsilon x}$.
Printed answer:
- $\log_\epsilon(\log_\epsilon x) + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: passes log(log(x))
Exercise XIX, problem 9, p. 233
Find $\ds\int \dfrac{5x+1}{x^2 +x-2}\, dx$.
Printed answer:
- $2\log_\epsilon(x - 1) + 3\log_\epsilon(x + 2) + C$.
verified: the printed answer passed a computed check
How it was checked
- integrate: PASS-ABS-CONVENTION 2*log(x - 1) + 3*log(x + 2)