Trigonometric Integral by Slick Substitution

by Pablo:

Here's an interesting integral that needs a trigonometric identity and then apply ordinary u-substitution.

At first sight, this integral does not seem like anything we've seen before. However, here we can use a trigonometric identity. This is a trick that you probably used when learned to solve limits involving trigonometric limits.

First of all, we remind ourselves of this basic identity:



If we multiply both sides by -1 and add 1:



If we now multiply both sides by two:



Now, of course, we make the substitution:



So, we end up with the identity:



Replacing this in our integral we get:



Now we use simple u-substitution, this is the easier part:



So, we have:



This is a nice trigonometric trick that is convenient to know.

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