An important class of integrals is of the form:

where m and n are integers. If either m or n is odd, factor out a single power of that function and rewrite the integral to solve with a u substitution. This is best demonstrated with an example:

Here, both m and n are odd, so we can select either function to factor. The process is easier when you pick the smaller of the two exponents, so let’s choose the sin x and we factor out one power as follows:

We then rewrite the even power as a power of sin^{2} x so that we can apply a trig identity:

This is easily integrated with a u substitution (let u = cos x):

So what do you do if both m and n are even? Well, most students just skip the problem and go on to the next one. ☺ But if you want to see the technique, look for my next blog post.