Queen's Rule: integral(0 to 2a) f(x) dx = integral(0 to a) [f(x) + f(2a-x)] dx

Case 1: f(2a-x) = f(x) => integral = 2 * integral(0 to a) f(x) dx Case 2: f(2a-x) = -f(x) => integral = 0

Application for [0, pi]: integral(0 to pi) x*f(sin x) dx. Apply King (x -> pi-x): I = integral(0 to pi) (pi-x)*f(sin(pi-x)) dx = integral(0 to pi) (pi-x)*f(sin x) dx 2I = pi * integral(0 to pi) f(sin x) dx = 2pi * integral$\frac{0 to pi}{2}$ f(sin x) dx

This reduces integrals of the form integral(0 to pi) x*g(sin x) dx to pi times a simpler integral.