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How to Integrate Sine/Cosine Problems with Even, Nonnegative Powers of Both Sine and Cosine

Here’s how you integrate a trig integral that contains sines and cosines where the powers of both sine and cosine are even and nonnegative (in other words, zero or positive). You first convert the integrand into odd powers of cosines by using the following trig identities:

Then you finish the problem as described below.

Here’s an example:

The first in this string of integrals is a no-brainer; the second is a simple reverse rule with a little tweak for the 2; you do the third integral by using the cosine squared identity (shown above) a second time; and the fourth integral is handled as follows:

You want to integrate a trig integral that contains sines and cosines where the power of cosine is odd and positive. So, you lop off one cosine factor and put it to the right of the rest of the expression, convert the remaining (even) cosine factors to sines with the Pythagorean identity, and then integrate with the substitution method where u = sin(x).