(Yes, I’m bored. So why not completing my series.)
First, remember that , so by chain rule we obtain . By the (first) fundamental theorem of calculus, derivative and antiderivative are two inverse operations, thus we obtain .
Also, recall that and . Don’t tell me you don’t know this; if you don’t, then stop reading.
To the real meat.
Letting , we have (!). Thus,
Which by our lemma above is equal to
(Hint: Wikipedia is your friend.)
Usually I stop here, but you can also simplify it to involve only one trigonometric function:
Call . So
Now, what happens to …