# Trigonometric Integrals

 Friday: Quiz 2 Next: Trig subst.

 and (5.2)

Example 5.2.1   Compute .
We use trig. identities and compute the integral directly as follows:

 (substitution )

This always works for odd powers of .

Example 5.2.2   What about even powers?! Compute . We have

Thus

Key Trick: Realize that we should write as . The rest is straightforward.

Example 5.2.3   This example illustrates a method for computing integrals of trig functions that doesn't require knowing any trig identities at all or any tricks. It is very tedious though. We compute using complex exponentials. We have

hence

The answer looks totally different, but is in fact the same function.

Here are some more identities that we'll use in illustrating some tricks below.

 and Also,

Example 5.2.4   Compute . We have

Here we used the substitution , so , so

Also, with the substitution and we get

Key trick: Write as .

Example 5.2.5   Here's one that combines trig identities with the funnest variant of integration by parts. Compute .
We have

Let's use integration by parts.

The above integral becomes

This is familiar. Solve for . We get

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William Stein 2006-03-15