Lecture 36 - Trigonometric Integrals

***MISTAKE IN VIDEO*** The Fourier series should run from n=0 to infinity, NOT n= -infinity to infinity when the series is in terms of sines and cosines. It runs from -infinity to infinity when expressed in terms of the exponential (e^{i n pi/L x}). As I say in the video, there's a magic way of changing the coefficients c_n in the exponential form into the a_n and b_n coefficients in the sin and cos form. When you do that, you end up combining c_1 with c_-1 to get an a_1 and b_1, c_2 with c_-2 to get a_2 and b_2, and so on. This changes the complex coefficients c_n and c_-n into real coefficients a_n and b_n. In this way, we only need to run n from n=0 to infinity. Doing this also ensures that cos(n pi/L x) and sin(n pi/L x) do not need to run through the negative values of n. Sorry! I might do a more in-depth video on how that's done in the future. - How to approach integrals involving powers of sines and cosines - A broad introduction to Fourier series