prove that integral sin(m*pi*x/L)*sin(n*pi*x/L)*dx from -L to L = {0, if m is not equal to n; L, if m = n} Solution if m=n sin(m*pi*x/L)*sin(n*pi*x/L)*dx = sin(m*pi*x/L)*sin(m*pi*x/L)*dx = sin^2 (m*pi*x/L) =1/2( 1+cos 2(m*pi*x/L) ) integration is | x/2 - L/4pi * sin2(m*pi*x/L) | -L to L = L + 0 = L.