∫sin^4x dx
=∫(1-cos^2x )sin^2xdx
=∫sin^2xdx-1/4∫(sin2x)^2dx
=1/2∫(1-cos2x)dx-1/8∫(1-cos4x)dx
=1/2x-1/2sin2x-1/8x+1/4sin4x+C
=3/8x-1/2sin2x+1/4sin4x+C