When is it appropriate to solve an integral with a trig substitution? First of all, keep in mind that a trig substitution doesn’t always work. Even when it does work, you are often left with an integral that will require other techniques such as a u substitution or integration by parts. But if you are willing to put in a little effort (and you know your trig identities), trig substitutions allow you to find the antiderivatives of some rather complicated functions.

There are three conditions that you look for—each a radical term of a particular form in the integrand. Each condition is associated with a different substitution. After you make the substitution, you simplify the integrand and go from there.

Term |
Substitution |
Radical becomes… |

Before we look at some example integrals, let’s see why the first radical term above simplifies to It’s pretty straightforward if you know your trig identities:

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Example. Integrate the following:

Solutions:

- (Does this integral look familiar?) Here, , so use Using the first line of the table above:

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But since

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- Here, and we use line 2 from the table above Note that Upon substitution,

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Hmm. This is going to take a little bit of extra work… Time to pull out some trig identities:

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Now a u substitution, letting :

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How do we get our answer back in terms of Draw a triangle that shows how and are related, then use the Pythagorean theorem to find an expression for In the triangle below,

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Substitute into the integral above to get:

This can be simplified further by factoring:

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Whew! - This is a trick question. Even though it fits the condition given in the table (and you could integrate with a trig substitution if you wanted), it’s easier to do this one with a u substitution: and

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The lesson here is to look for u substitutions before you look for trig substitutions.