he indefinite integral can be found by substitution:u=sin(x)du = cos(x)dxand use sin^2(x)+cos^2(x)=1 to convert sin^4(x)cos^4(x)=sin^4(x)(1-sin^2(x))^2=u^4(1-u^2)^2SoI=integral(u^4(1-u^2)^2duAnd you'll soon find that the integral with limits -pi to pi, -pi to 0, 0 to pi all give 0.
To make it more understandable,∫sin4(x)cos5(x)dx = ∫u4(1−u2)2du
withu=sin(x), du=cos(x)dx