# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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Symmetry For The Win. Topic: Calculus. October 30th, 2006

Problem: Evaluate the improper integral .

Solution: As the topic suggests, we will look for a symmetry to simplify the problem. Notice the identity (since we’re just integrating across the interval from different “directions.” Using this, we have

.

Adding the old integral to the new one, we have

from the property of logs and the double-angle identity (here it is again!). But in fact this expression is simply

(*)

by the substitution . Taking into the account of the symmetry of from to , we get so

.

Plugging back into (*) we obtain so we get . QED.

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Comment: The function is not nice to actually integrate; it involves the polylogarithm function if you try here. This is an important example of how symmetry can help a ton in integration because there are so many functions that cannot be integrated with elementary functions but can be evaluated over an interval through different techniques.

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Practice Problem: Evaluate without actually finding the antiderivative.