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:

if . (The cases and have already been handled earlier in this series.)

To complete the calculation, I employ the now-familiar antiderivative

.

Using this antiderivative and a simple substitution, I see that

.

This completes the fourth method of evaluating the integral , using partial fractions.

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