Use the previous four exercises (Exercise #7, Exercise #8, Exercise #9, and Exercise #10) to prove the following:
- .
- .
- .
- Proof. First, to apply integration by parts we define the following functions
We used Exercise #7 for the primitive of . So, we then have
- Proof. To apply integration by parts we define the following functions
Where we obtained the primitive of from Exercise #10. Then applying integration by parts we have
- Proof. To apply integration by parts we define the following functions
Where we derived the primitive for in part (a) above. So, using integration by parts we have
Then, we need to use integration by parts again to evaluate . Let
Integrating by parts
Now, plugging this back in where we left off in evaluating we have