* 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

Copyright 2005 by Bruce Ikenaga