Let
- Find all values of such that .
- Prove that
- Similar to part (b), find expressions for
- Recalling the definition of as the integral,
We compute,
Then, since
we have
and
Hence, the set of such that is all real .
- Proof. We start by computing,
Then, using the translation invariance of the integral,
- We claim the following formulas hold:
Proof. Making the substitution , we have
Proof. To prove this identity we integrate by parts, letting
Therefore,
Proof. For the final identity, we use the substitution , . This gives us
Now, we use integration by parts with
This gives us