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:

,

where

,

,

.

I’ll begin with and apply the substitution , or . Then , and the endpoints change from to . Therefore,

.

Next, we use the periodic property for both sine and cosine — and — to rewrite as

.

Changing the dummy variable from back to , we have

.

Therefore, we can combined into a single integral:

Next, we work on the middle integral . We use the substitution , or , so that . Then the interval of integration changes from to , so that

.

Next, we use the trigonometric identities

,

,

so that the last integral becomes

On the line above, I again replaced the dummy variable of integration from to . We see that , and so

I’ll continue with the evaluation of this integral in tomorrow’s post.

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