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!