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Thursday, November 17, 2011

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.

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