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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