8.4: Trigonometric Substitutions

So far we have seen that it sometimes helps to replace a subexpression of a function by a single variable. Occasionally it can help to replace the original variable by something more complicated. This seems like a "reverse'' substitution, but it is really no different in principle than ordinary substitution.

Example $\PageIndex{1}$

Evaluate

$$\int \sqrt{1-x^2}\,dx.$$

Solution

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 $\sqrt{\cos^2 u}$ by $\cos u$, but this is valid only if $\cos u$ is positive, since $\sqrt{\cos^2 u}$ 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

$$\sin 2u=2\sin u\cos u=2\sin(\arcsin x)\sqrt{1-\sin^2 u}= 2x\sqrt{1-\sin^2(\arcsin x)}=2x\sqrt{1-x^2}.$$

Then the full antiderivative is

$${\arcsin x\over2}+{2x\sqrt{1-x^2}\over4}= {\arcsin x\over2}+{x\sqrt{1-x^2}\over2}+C.$$

This type of substitution is usually indicated when the function you wish to integrate contains a polynomial expression that might allow you to use the fundamental identity $\sin^2x+\cos^2x=1$ in one of three forms:

$$\cos^2 x=1-\sin^2x,$$

$$\sec^2x=1+\tan^2x,$$

or

$$\tan^2x=\sec^2x-1.$$

If your function contains $1-x^2$, as in the example above, try $x=\sin u$; if it contains $1+x^2$ try $x=\tan u$; and if it contains $x^2-1$, try $x=\sec u$. Sometimes you will need to try something a bit different to handle constants other than one.

Example $\PageIndex{2}$

Evaluate $$\int\sqrt{4-9x^2}\,dx.$$

Solution

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 fromExample$\PageIndex{1}$,

Example $\PageIndex{3}$

Evaluate $$\int\sqrt{1+x^2}\,dx.$$

Solution

Let $x=\tan u$, $dx=\sec^2 u\,du$, 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 $\sqrt{\sec^2u}=\sec u$. Then $$\int\sqrt{\sec^2u}\sec^2u\,du=\int \sec^3 u \,du.$$ In problems of this type, two integrals come up frequently: $\int\sec^3u\,du$ and $\int\sec u\,du$. 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 $\int\sec^3u\,du$:

$$\eqalign{ \int\sec u\,du&=\int\sec u\,{\sec u +\tan u\over \sec u +\tan u}\,du\cr &=\int{\sec^2 u +\sec u\tan u\over \sec u +\tan u}\,du.\cr }$$

Now let $w=\sec u +\tan u$, $dw=\sec u \tan u + \sec^2u\,du$, 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 $\int\sec^3 u\,du$:

$$\eqalign{ \sec^3u&={\sec^3u\over2}+{\sec^3u\over2}={\sec^3u\over2}+{(\tan^2u+1)\sec u\over 2}\cr &={\sec^3u\over2}+{\sec u \tan^2 u\over2}+{\sec u\over 2}= {\sec^3u+\sec u \tan^2u\over 2}+{\sec u\over 2}.\cr }$$

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 $\sec(\arctan x)=\sqrt{1+\tan^2(\arctan x)}=\sqrt{1+x^2}$.

Contributors

• David Guichard (Whitman College)

• Integrated by Justin Marshall.