Example 6.3.1. Integrating powers of sine and cosine.
Evaluate
Solution 1.
But what if, for some reason, we wanted to let instead? Unfortunately, we have as part of our integrand, not just The solution to this problem is to replace some of our powers of sine (two of them to be exact) with expressions that involve cosine. We will use the Pythagorean Identity
This looks like a very different answer, so you might wonder if we went wrong somewhere. But in fact, the two answers are equivalent, in the sense that they differ by a constant! (So the “ ” is different in each case, if you like.) Notice that
so the difference between the two answers is the constant .