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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