# 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 Subsections
William Stein 2006-03-15