Previously in this series, I showed that

So far, I have shown that

where and (and is a certain angle that is now irrelevant at this point in the calculation).

There are actually a couple of ways for computing this last integral. Today, I’ll lay the foundation for the “magic substitution”

With this substitution, the above integral will become a rational function, which can then be found using standard techniques.

First, we use some trig identities to rewrite in terms of :

Next, I’ll replace by :

.

Second, for the sake of completeness (even though it isn’t necessary for this particular integral), I’ll rewrite in terms of :

Next, I’ll replace by :

.

Third, again for the sake of completeness,

.

Finally, I need to worry about what happens to the :

These four substitutions can be used to convert trigonometric integrals into some other integral. Usually, the new integrand is pretty messy, and so these substitutions should only be used sparingly, as a last resort.

I’ll continue this different method of evaluating this integral in tomorrow’s post.