How to prove that these statements are correct....?

I'm uming you are able to integrate these expressions, and that you are wanting to know how to prove that the result is not zero for m=n. Now, do not set any of you sin(n*pi), or similar terms, to zero; just keep them. Then notice that you have a (m-n) term in the denominator which gives you a divide by zero (0/0 is an indeterminate form so take care with this case), so use L'hopitals rule to handle this case seperately and the answer should come right out. I'm also uming you know how to use L'hopitals rule... let me know if you need more information or if I need to clarify anything.