Compute the following integral:

We have

On the second line we used the theorem on expansion/contraction of the interval of integration. Theorem 1.19 in Apostol, I believe.

Note, on the first line we used the trig identity . I keep evaluating the integrals this way. I think Apostol does give a formula for the solution of in the text, so you could save some effort by using that. I can’t always remember all those specialized formulas, so it’s easier to just solve it directly from the trig identity, since I remember those and they come up all the time in other contexts. Of course, as always, do what you like best.

*Related*

I think that the sine should be -sin(2y)/4, because when you used the expansion contraction theorem you forgot to multiply the upper limit.