# Trig Substitution

Trig substitution reduces certain integrals to integrals of trig functions. The idea is to match the given integral against one of the following trig identities:

• If the integral contains an expression of the form , try a substitution based on the first identity: .
• If the integral contains an expression of the form , try a substitution based on the second identity: .
• If the integral contains an expression of the form , try a substitution based on the third identity: .

If you don't obtain one of the identities above after substituting, you've probably used the wrong substitution.

Example.

To "match" the "4" in " ", I had to use (since ). I used the double angle formula to reduce the even powers of cosine.

To put the x's back, I need to express everything in terms of trig functions of (as opposed to or ). I use the double angle formulas for sine:

Therefore,

Now draw a right triangle which shows the substitution.

The triangle shows , and by Pythagoras the third side is . Therefore,

Example. Compute .

looks like , so let . Then , so

Example. Compute .

This could be done using . But it's easier to do a u-substitution:

Example. Compute .

looks like , so let . Then , and

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