cos 2x = 1 - 2sin^2 x = 2cos^2 x - 1
sin^2 x = (cos 2x - 1)/2
cos^2 x = (1 + cos 2x)/2
∫ sin^4 x dx = ∫(cos 2x - 1)^2/4 dx =
= 1/4*∫ (cos^2 2x - 2cos 2x + 1) dx =
= 1/4*∫ (1/2*(1 + cos 4x) - 2cos 2x + 1) dx =
= 1/8*x + 1/8*1/4*sin 4x - 1/4*2/2*sin 2x + 1/4*x + С =
= 3/8*x + 1/32*sin 4x - 1/4*sin 2x + С