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:

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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