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.
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.
Practice Problem: Evaluate without actually finding the antiderivative.
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