Find a formula to compute

for all for the following function .

- .
- The function,
- .
- the maximum of 1 and .

- We know from this exercise (Section 5.5, Exercise #13) that
- If , then over the whole integral, and so
Then, if we have

(Since we have so this equation works. This is the form Apostol wrote these answers as in the back of the book, so I’m getting our answers to match his. I wouldn’t have written them this way otherwise.)

Finally, if we have

Since the formulas for are the same for and are the same we have

for .

- We consider two cases. If then
If then

- Since the maximum of 1 and is equal to 1 if and is equal to if we consider three cases (, and ).
For we have

For we have

For we have

I think (b) is wrong. You cannot simply convert into . The second term for the negative case (< -1) should be . The answer in Apostol is wrong, which is easy to check if you plot it.