Integrating as the Reverse of Differentiating

Excerpts

• Integrating as the Reverse of Differentiating

  So, therefore, when we reverse the process we must always remember to add on this undetermined constant, even if we do not yet know what its value will be.

• Integrating as the Reverse of Differentiating

  *N.B.*---Here note this very remarkable fact, that we could not have integrated in the above case if we had not happened to know the corresponding differentiation.

• Integrating as the Reverse of Differentiating

  Clearly, in dealing with powers of $x$, the rule for working backwards will be: Increase the power by $1$, then divide by that increased power, and add the undetermined constant.

• Integrating as the Reverse of Differentiating

  Hence, when you work the other way and integrate, the constant reappears multiplied by $x$.

• Integrating as the Reverse of Differentiating

  Thus, if $\dfrac{dy}{dx} = 4x^2$, the reverse process gives us $y = \frac{4}{3}x^3$.

• Integrating as the Reverse of Differentiating

  But here comes in a curious point. We should get $\dfrac{dy}{dx} = 4x^3$ if we had begun with *any* of the following: $x^4$, or $x^4 + a$, or $x^4 + c$, or $x^4$ with *any* added constant.

• Integrating as the Reverse of Differentiating

  We haven’t yet integrated: we have only written down instructions to integrate---if we can. Let us try. Plenty of other fools can do it---why not we also?

• Integrating as the Reverse of Differentiating

  But when we come to the right-hand side of the equation we must remember that what we have got to sum up together is not all the $dx$’s, but all such terms as $x^2\, dx$; and this will *not* be the same as $x^2 \ds\int dx$, because $x^2$ is not a constant.

• Integrating as the Reverse of Differentiating

  If a stranger were set down in Trafalgar Square, and told to find his way to Euston Station, he might find the task hopeless. But if he had previously been personally conducted from Euston Station to Trafalgar Square, it would be comparatively easy to him to find his way back to Euston Station.

• Integrating as the Reverse of Differentiating

  So, when we work backwards, integrating, the integration will be simply the sum of the two separate integrations.

• Integrating as the Reverse of Differentiating

  Then, of course, since we know that differentiating $\log_\epsilon x$ gives us $x^{-1}$, we know that, by reversing the process, integrating $dy = x^{-1}\, dx$ will give us $y = \log_\epsilon x$.

• Integrating as the Reverse of Differentiating

  Indeed it should be frankly admitted that this is one of the curious features of the integral calculus:---that you can’t integrate anything before the reverse process of differentiating something else has yielded that expression which you want to integrate.

• Integrating as the Reverse of Differentiating

  You should make such a table for yourself, putting in it only the general functions which you have successfully differentiated and integrated. See to it that it grows steadily!

Equations

• Integrating as the Reverse of Differentiating

  y = \frac{1}{n + 1} x^{n+1} + C

  Working backwards from dy/dx = x^n gives y equal to x raised to n+1, divided by n+1, plus an undetermined constant.

• Integrating as the Reverse of Differentiating

  \int x^n\, dx = \dfrac{1}{n+1} x^{n+1}

  The integral of x to the power n with respect to x is x to the power n+1 divided by n+1.

• Integrating as the Reverse of Differentiating

  \frac{dy}{dx} = anx^{n-1}

  Differentiating y = a x^n gives a times n times x to the power n-1, the constant factor a carried through.

• Integrating as the Reverse of Differentiating

  y = \frac{1}{n+1} x^{n+1} + bx + C

  Integrating dy/dx = x^n + b gives x to the power n+1 over n+1, plus b times x, plus a constant of integration.

• Integrating as the Reverse of Differentiating

  \int x^{-1}\, dx = \log_\epsilon x + C

  The integral of 1/x is the natural logarithm of x plus a constant; this is the exceptional case the power rule does not cover.

• Integrating as the Reverse of Differentiating

  \int \frac{1}{x+a}\, dx = \log_\epsilon (x+a) + C

  The integral of 1/(x+a) with respect to x is the natural logarithm of x+a plus a constant.

• Integrating as the Reverse of Differentiating

  \int \epsilon^x\, dx = \epsilon ^x + C

  The exponential function e^x is its own integral, up to a constant.

• Integrating as the Reverse of Differentiating

  \int \epsilon^{-x}\, dx = -\epsilon^{-x} + C

  The integral of e^(-x) with respect to x is minus e^(-x), plus a constant.

• Integrating as the Reverse of Differentiating

  \int \sin x\, dx = -\cos x + C

  The integral of sin x is minus cos x, plus a constant.

• Integrating as the Reverse of Differentiating

  \int \cos x\, dx = \sin x + C

  The integral of cos x is sin x, plus a constant.

• Integrating as the Reverse of Differentiating

  \int\log_\epsilon x\, dx = x(\log_\epsilon x - 1) + C

  The integral of the natural logarithm of x is x times (the natural logarithm of x minus 1), plus a constant.

• Integrating as the Reverse of Differentiating

  \int\log_{10} x\, dx = 0.4343x (\log_\epsilon x - 1) + C

  The integral of the common logarithm of x is about 0.4343 times x times (the natural logarithm of x minus 1), plus a constant; 0.4343 is the book's rounded value of 1/ln 10.

• Integrating as the Reverse of Differentiating

  \int a^x\, dx = \dfrac{a^x}{\log_\epsilon a} + C

  The integral of a to the power x is a to the power x divided by the natural logarithm of a, plus a constant.

• Integrating as the Reverse of Differentiating

  \int\cos ax\, dx = \frac{1}{a} \sin ax + C

  The integral of cos(ax) with respect to x is sin(ax) divided by a, plus a constant.

• Integrating as the Reverse of Differentiating

  \int\sin ax\, dx = -\frac{1}{a} \cos ax + C

  The integral of sin(ax) with respect to x is minus cos(ax) divided by a, plus a constant.

• Integrating as the Reverse of Differentiating

  \cos 2\theta = \cos^2\theta - \sin^2\theta

  The cosine of twice an angle equals the square of its cosine minus the square of its sine.

• Integrating as the Reverse of Differentiating

  \int x^n\, dx = \dfrac{1}{n+1} x^{n+1}.

  Integrating x to the power n gives x to the power n+1 divided by n+1 (the constant of integration is added separately).

• Integrating as the Reverse of Differentiating

  y = a \log_\epsilon x + C.

  The integral of a^... type constant-multiplied x^-1 dx is a times the natural logarithm of x, plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \int x^{-1}\, dx &&= \log_\epsilon x + C.

  The integral of 1/x with respect to x is the natural logarithm of x plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \int \epsilon^x\, dx &&= \epsilon ^x + C.

  The integral of Euler's number to the power x is itself, plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \int \sin x\, dx &&= -\cos x + C.

  The integral of sin x with respect to x is minus cos x, plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \int \cos x\, dx &&= \sin x + C.

  The integral of cos x with respect to x is sin x, plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \int\log_\epsilon x\, dx &&= x(\log_\epsilon x - 1) + C

  The integral of the natural logarithm of x is x times (log x minus 1), plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \int a^x\, dx &&= \dfrac{a^x}{\log_\epsilon a} + C.

  The integral of a to the power x is a to the power x divided by the natural logarithm of a, plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \int\cos ax\, dx &&= \frac{1}{a} \sin ax + C

  The integral of cos(ax) with respect to x is sin(ax) divided by a, plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \int\sin ax\, dx &&= -\frac{1}{a} \cos ax + C.

  The integral of sin(ax) with respect to x is minus cos(ax) divided by a, plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \int\log_{10} x\, dx &&= 0.4343x (\log_\epsilon x - 1) + C.

  The integral of the base-10 logarithm of x is about 0.4343 times x times (natural log of x minus 1), plus the constant of integration.

• Integrating as the Reverse of Differentiating

  \text{volume} = \iiint f(x,y,z) · dx · dy · dz.

  The volume of a solid is the triple integral of the function f over the small cubes dx dy dz filling the solid.

Problems