Example 8.3.1 Evaluate $\ds\int \sqrt{1-x^2}\,dx$. Let $x=\sin u$ so $dx=\cos u\,du$. Then $$ \int \sqrt{1-x^2}\,dx=\int\sqrt{1-\sin^2 u}\cos u\,du= \int\sqrt{\cos^2 u}\cos u\,du. $$ We would like to replace $\ds \sqrt{\cos^2 u}$ by $\cos u$, but this is valid only if $\cos u $ is positive, since $\ds \sqrt{\cos^2 u}$ is positive. Consider again the substitution $x=\sin u$. We could just as well think of this as $u=\arcsin x$. If we do, then by the definition of the arcsine, $-\pi/2\le u\le\pi/2$, so $\cos u\ge0$. Then we continue: $$\eqalign{ \int\sqrt{\cos^2 u}\cos u\,du&=\int\cos^2u\,du=\int {1+\cos 2u\over2}\,du = {u\over 2}+{\sin 2u\over4}+C\cr &={\arcsin x\over2}+{\sin(2\arcsin x)\over4}+C.\cr }$$ This is a perfectly good answer, though the term $\sin(2\arcsin x)$ is a bit unpleasant. It is possible to simplify this. Using the identity $\sin 2x=2\sin x\cos x$, we can write $\ds \sin 2u=2\sin u\cos u=2\sin(\arcsin x)\sqrt{1-\sin^2 u}= 2x\sqrt{1-\sin^2(\arcsin x)}=2x\sqrt{1-x^2}.$ Then the full antiderivative is $$ {\arcsin x\over2}+{2x\sqrt{1-x^2}\over4}= {\arcsin x\over2}+{x\sqrt{1-x^2}\over2}+C. $$ $\square$

Example 8.3.2 Evaluate $\ds\int\sqrt{4-9x^2}\,dx$. We start by rewriting this so that it looks more like the previous example: $$ \int\sqrt{4-9x^2}\,dx=\int\sqrt{4(1-(3x/2)^2)}\,dx =\int 2\sqrt{1-(3x/2)^2}\,dx. $$ Now let $3x/2=\sin u$ so $(3/2)\,dx=\cos u \,du$ or $dx=(2/3)\cos u\,du$. Then $$\eqalign{ \int 2\sqrt{1-(3x/2)^2}\,dx&=\int 2\sqrt{1-\sin^2u}\,(2/3)\cos u\,du ={4\over3}\int \cos^2u\,du\cr &={4u\over 6}+{4\sin 2u\over12}+C\cr &={2\arcsin(3x/2)\over3}+{2\sin u \cos u\over3}+C\cr &={2\arcsin(3x/2)\over3}+{2\sin(\arcsin(3x/2))\cos(\arcsin(3x/2))\over3}+C\cr &={2\arcsin(3x/2)\over3}+{2(3x/2)\sqrt{1-(3x/2)^2}\over3}+C\cr &={2\arcsin(3x/2)\over3}+{x\sqrt{4-9x^2}\over2}+C,\cr }$$ using some of the work from example 8.3.1. $\square$

Example 8.3.3 Evaluate $\ds\int\sqrt{1+x^2}\,dx$. Let $x=\tan u$, $\ds dx=\sec^2 u\,du$, so $$ \int\sqrt{1+x^2}\,dx=\int \sqrt{1+\tan^2 u}\sec^2u\,du= \int\sqrt{\sec^2u}\sec^2u\,du. $$ Since $u=\arctan(x)$, $-\pi/2\le u\le\pi/2$ and $\sec u\ge0$, so $\ds \sqrt{\sec^2u}=\sec u$. Then $$\int\sqrt{\sec^2u}\sec^2u\,du=\int \sec^3 u \,du.$$ In problems of this type, two integrals come up frequently: $\ds \int\sec^3u\,du$ and $\ds\int\sec u\,du$. Both have relatively nice expressions but they are a bit tricky to discover.

First we do $\int\sec u\,du$, which we will need to compute $\ds \int\sec^3u\,du$: $$\eqalign{ \int\sec u\,du&=\int\sec u\,{\sec u +\tan u\over \sec u +\tan u}\,du\cr &=\int{\sec^2 u +\sec u\tan u\over \sec u +\tan u}\,du.\cr }$$ Now let $w=\sec u +\tan u$, $\ds dw=\sec u \tan u + \sec^2u\,du$, exactly the numerator of the function we are integrating. Thus $$\eqalign{ \int\sec u\,du=\int{\sec^2 u +\sec u\tan u\over \sec u +\tan u}\,du&= \int{1\over w}\,dw=\ln |w|+C\cr &=\ln|\sec u +\tan u|+C.\cr }$$

Now for $\ds \int\sec^3 u\,du$: $$\eqalign{ \sec^3u&={\sec^3u\over2}+{\sec^3u\over2}={\sec^3u\over2}+{(\tan^2u+1)\sec u\over 2}\cr &={\sec^3u\over2}+{\sec u \tan^2 u\over2}+{\sec u\over 2}= {\sec^3u+\sec u \tan^2u\over 2}+{\sec u\over 2}.\cr }$$ We already know how to integrate $\sec u$, so we just need the first quotient. This is "simply'' a matter of recognizing the product rule in action: $$\int \sec^3u+\sec u \tan^2u\,du=\sec u \tan u.$$

So putting these together we get $$ \int\sec^3u\,du={\sec u \tan u\over2}+{\ln|\sec u +\tan u| \over2}+C, $$ and reverting to the original variable $x$: $$\eqalign{ \int\sqrt{1+x^2}\,dx&={\sec u \tan u\over2}+{\ln|\sec u +\tan u|\over2}+C\cr &={\sec(\arctan x) \tan(\arctan x)\over2} +{\ln|\sec(\arctan x) +\tan(\arctan x)|\over2}+C\cr &={ x\sqrt{1+x^2}\over2} +{\ln|\sqrt{1+x^2} +x|\over2}+C,\cr }$$ using $\tan(\arctan x)=x$ and $\ds \sec(\arctan x)=\sqrt{1+\tan^2(\arctan x)}=\sqrt{1+x^2}$. $\square$