Earlier in this series, I gave three different methods of showing that

Using the fact that is independent of , I’ll now give a fourth method.

Since is independent of , I can substitute any convenient value of that I want without changing the value of . As shown in previous posts, substituting yields the following simplification:

Since is independent of , I can substitute any convenient value of that I want without changing the value of . As shown in previous posts, substituting yields the following simplification:

I now employ the magic substitution , so that

,

,

.

The endpoints change from to , and so

I have transformed the integral into a new integral involving a fairly simple rational function that can be evaluated using standard (and non-standard) techniques.

Hypothetically, the magic substitution can be applied to the original integral. Unfortunately, I was unable to make any headway in finding the four complex roots of the resulting rational function. However, since I made the replacement at the start, this new rational function is much more tractable.

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