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:

To evaluate this integral, I need to find the four complex roots of the denominator:

To solve for , there are three separate cases that have to be considered: , , and . I’ll begin with the easiest case of . In this case, the integral is easy to evaluate:

This matches the expected answer of since I used the assumption that .

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