Trigonometric functions can be substituted for other expressions to change the form of integrands. One may use the trigonometric identities to simplify certain integrals containing radical expressions (or expressions containing $n$th roots). The following are general methods of trigonometric substitution, depending on the form of the function to be integrated.

Substitution Rule #1

If the integral contains $a^2-x^2$, let $x = a \sin(\theta)$ and use the identity: 

$1-\sin^2(\theta) = \cos^2(\theta)$

Substitution Rule #2

If the integrand contains $a^2+x^2$, let $x = a \tan(\theta)$ and use the identity:

$1+\tan^2(\theta) = \sec^2(\theta)$

Substitution Rule #3

If the integrand contains $x^2-a^2$, let $x = a \sec(\theta)$ and use the identity:

$\sec^2(\theta)-1 = \tan^2(\theta)$

Note that, for a definite integral, one must figure out how the bounds of integration change due to the substitution.

Examples

In order to better understand these substitutions, let's go over the derivation of some of them.

Example 1: Integrals where the integrand contains $a^2 − x^2$ (where $a$ is positive)

In the integral 

$\displaystyle{\int\frac{dx}{\sqrt{a^2-x^2}}}$

we may use:

 $\displaystyle{x=a\sin(\theta)\\ dx=a\cos(\theta)\,d\theta\\ \theta=\arcsin\left(\frac{x}{a}\right)}$

With the substitution, we get:

$\begin{aligned} \int\frac{dx}{\sqrt{a^2-x^2}} & = \int\frac{a\cos(\theta)\,d\theta}{\sqrt{a^2-a^2\sin^2(\theta)}} \\ &= \int\frac{a\cos(\theta)\,d\theta}{\sqrt{a^2(1-\sin^2(\theta))}} \\ &= \int\frac{a\cos(\theta)\,d\theta}{\sqrt{a^2\cos^2(\theta)}} \\ &= \int d\theta \\ &= \theta+C \\ &= \displaystyle{\arcsin \left(\frac{x}{a}\right)}+C \end{aligned}$

Example 2: Integrals where the integrand contains $a^2 − x^2$ (where $a$ is not zero)

In the integral 

$\displaystyle{\int\frac{dx}{{a^2+x^2}}}$

we may use:

 $\displaystyle{x=a\tan(\theta)\\ dx=a\sec^2(\theta)\,d\theta\\ \theta=\arctan\left(\frac{x}{a}\right)}$

With the substitution, we get:

$\begin{aligned}\int\frac{dx}{{a^2+x^2}} &= \int\frac{a\sec^2(\theta)\,d\theta}{{a^2+a^2\tan^2(\theta)}} \\ &= \int\frac{a\sec^2(\theta)\,d\theta}{{a^2(1+\tan^2(\theta))}} \\ &= \int \frac{a\sec^2(\theta)\,d\theta}{{a^2\sec^2(\theta)}} \\ &= \int \frac{d\theta}{a} \\ &= \frac{\theta}{a}+C \\ &= \frac{1}{a} \arctan \left(\frac{x}{a}\right)+C\end{aligned}$