Wednesday, November 16th

Problem
Let $I_m=\int_0^{2\pi}\cos(x )\cos(2s)\dots\cos(mx)dx$. For which integers &1\le m\le 10$ is $I_m\neq 0$?

Solution
Since $\cos(kx)$ is a $k$ degree even or odd polynomial in $\cos(x)$ whether $k$ is even or odd respectively. Since the integral of an odd power of $\cos(x)$ is zero and the integral of an even power is nonzero, we need the integral in $I_m$ to be an even polynomial. Therefore the values that satisfy $I_m\neq0$ are $m=3,4,7,8$.