∫0 to π/2 sin^2(x) cos^3(x) dx = ∫0 to π/2 sin^2(x) cos^2(x) cos(x) dx = ∫0 to π/2 sin^2(x) (1 - sin^2(x)) cos(x) dx. Let u = sin(x), then du = cos(x) dx. When x = 0, u = 0. When x = π/2, u = 1. So, the integral becomes ∫0 to 1 u^2 (1 - u^2) du = ∫0 to 1 (u^2 - u^4) du = [u^3/3 - u^5/5] from 0 to 1 = (1/3 - 1/5) - (0 - 0) = (5 - 3)/15 = 2/15.