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.
Return to Integration by SubstitutionReturn to Home Page