Well, I never really discussed these kind of things in front of my classmates. My teachers just got sick of me asking questions before and after a lesson, and during lunch. So now I just look up what I want on the net.
Now back to Fus Ro Dah's question, would the value of the integral be the same if we shifted the upper and lower bounds of the integral by +/- k(pi/2)? It should be the same shouldn't it? Looking at the graph of cos^2m(theta), this seems to be the case.