**Integration by trigonometric substitution** is a technique of integration that involves substituting some function of x for a trigonometric function.

As a general rule, when taking an antiderivative of a function in the form , the substitution is usually the best option. For and , the substitutions and (respectively) are usually the best options.

## Examples

This technique can be used when functions would be otherwise difficult to integrate. One of the most well-known examples is

Here, we can use the substitution to get

Therefore:

A second example:

Here, we can use the substitution to get

By using the trigonometric identity , we get

Which evaluates to by using u-substitution. Since

we can say that

Therefore: