Previously in this series, I showed that
We now employ the substitution , so that . Also, the limits of integration change from to , so that
Next, I’ll divide write by dividing the interval of integration (not to be confused with the and used in the previous method), where
For , I employ the substitution , so that and . Under this substitution, the interval of integration changes from $2\pi \le \theta \le 4\pi$ to $0 \le u \le 2\pi$, and so
Next, I use the periodic property for both sine and cosine — and — to rewrite as
Except for the dummy variable , instead of , we see that is identical to . Therefore,
I’ll continue this different method of evaluating this integral in tomorrow’s post.