Establish that the following integration formula is correct:
Proof. First, we want to integrate by parts. Let
Therefore, we have
Now, to evaluate this integral we break it into pieces (to get rid of the ). Since we have
(Since is only defined for we don’t need to worry about the cases .) We then evaluate the two pieces of the integral separately. If ,
Now, we note that
Therefore, we have an integral of the form and so,
So, for we have
For the case everything is identical except we have a negative sign (since has a negative sign when ) so for we have
Therefore (since if and if ) we have
Brilliant!