What did you come up with?
Find an expression for in terms of m and s, where m is a positive rational number, m != -1 and s > 1. Show that the infinte integral has a meaning if m > 1, and state its value in terms of m.
Is it asking me to integrate and come up with a simplified expression? I did that, but it's not matching the answer on multiple attempts.
Re-write as , and then use the usual integration rule: Increase the power by 1, and then divide by the number you just thought of.
When you increase by you get , so let's see what it looks like:
(Note that this is where we need , because if , we'd be violating the first commandment of arithmetic: Thou shalt not divide by zero.)
Now what you have to do:
- Return to the denominator of the fraction, by changing the sign of its power.
- Put in the limits and being careful with the minus signs.
This should give you the answer to the first part:
Now you need to look at what happens when and . So, think about the sign of , and what happens to as if is positive, and if is negative.
Can you see what to do now?