Some husbands try to impress their wives by lifting extremely heavy objects or other extraordinary feats of physical prowess.

That will never happen in the Quintanilla household in a million years.

But she was impressed that I broke an impasse in her research and resolved a discrepancy between Mathematica 4 and Mathematica 8 by finding the following integral by hand in less than an hour:

In this series, I’ll explore different ways of evaluating this integral.So far in this series, I’ve shown that

This last integral can be evaluated using a standard trick. Let , so that . We differentiate this last equation with respect to :

Employing a Pythagorean identity, we have

Since , we may rewrite this as

Integrating both sides with respect to , we obtain the antiderivative

We now employ this antiderivative to evaluate :

And so, at long last, we have arrived at the solution for the integral . Surprisingly, the answer is independent of the parameter .

These last few posts illustrated the technique that I used to compute this integral for my wife in support of her recent paper in Physical Review A. However, I had more than a few false starts along the way… or, at the time, I thought they were false starts. It turns out that there are multiple ways of evaluating this integral, and I’ll explore another method of attack beginning with tomorrow’s post.

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