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:

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.

There’s at least one more way that the integral can be calculated, which I’ll begin with tomorrow’s post.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
View all posts by John Quintanilla

## One thought on “How I Impressed My Wife: Part 5j”