Prove the following integral formulas.
- .
- .
- .
- Guess and prove a general formula based on parts (a) – (c).
- Proof. We use integration by parts with
This gives us
- Proof. We use integration by parts and the result of part (a). Let
This gives us
- Proof. Again, we use integration by parts, and this time part (b). Let
This gives us
- Claim:
Proof. The proof is by induction. We have already established the case (and and ). Assume then that the formula holds for some positive integer . We then consider the integral
Integrating by parts, we let
Therefore, integrating by parts and using the induction hypothesis we have,
Therefore, the formula holds for , and hence, for all positive integers