∫ x^2sin^2x dx
= ∫ x^2*(1 - cos2x)/2 dx
= (1/2)∫ x^2 dx - (1/2)∫ x^2cos2x dx
= x^3/6 - (1/4)∫ x^2 d(sin2x)
= x^3/6 - (1/4)x^2sin2x + (1/4)∫ 2xsin2x dx
= x^3/6 - (1/4)x^2sin2x - (1/4)∫ x d(cos2x)
= x^3/6 - (1/4)x^2sin2x - (1/4)xcos2x + (1/4)∫ cos2x dx
= x^3/6 - (1/4)x^2sin2x - (1/4)xcos2x + (1/8)sin2x + C