∫[0,π/2]sin^8(x)dx = [-cos(x)*sin^7(x)][0,π/2] + 7∫[0,π/2]cos^2(x)*sin^6(x)dx
= 7∫[0,π/2]sin^6(x)dx - 7∫[0,π/2]sin^8(x)dx
∫[0,π/2]sin^8(x)dx = (7/8)∫[0,π/2]sin^6(x)dx

一般に　∫[0,π/2]sin^n(x)dx = {(n-1)/n}∫[0,π/2]sin^(n-2)(x)dx

与式 = (1/2)∫[0,π/2]sin^8(x)dx
= (1/2)*(7/8)*(5/6)*(3/4)*(1/2)*∫[0,π/2]dx
= (35/512)π