∫cos^3 x dx
=∫cosxcos^2xdx
=∫cosx(1-sin^2)dx
=∫cosxdx-∫cosxsin^2xdx
=-sinx-1/2∫sin2xsinxdx
=-sinx-1/2∫(-1/2(cos(3x)-cosx)dx
=-sinx+1/4∫cos3xdx-1/4∫cosxdx
=-sinx+1/4*1/3∫cos3xd3x+1/4sinx
=-3/4sinx-1/12sin3x+c