It's a bit tricky, but here we go:
Now we use integration by parts with and . So we get
Using the identity , we get
The integral on the RHS we recognize as so we get
Well the rest is very easy, just solve for and you should have (a).
Putting it to the test for your given integral, we need expressed in . Some basic manipulation should get you
Now your asked integral is basically , with boundaries filled in. You should be able to do the rest on your own. I can confirm that the answer is indeed .