Can somebody please explain to me why the integral of, for instance, cos((2*pi*x)/a)*cos((4*pi*x)/a) vanishes over the interval 0 to a? As I understand it, this is generally the case when integrating sines and cosines with different arguments "over the interval of a period." But I'm confused about why this is the case. Is there a straightforward argument. (I know it can be derived using complex exponentials, but I'm wondering if there is some kind of intuitive understanding of the graphs, like maybe using the properties of even and odd functions?) Also I'm somewhat confused about what it means here to "integrate over a period," since the interval from 0 to a, in the above example, is technically the full period of the first cosine, but only half that of the second, right? Any help is greatly appreciated. Thank you.