∫sin^6x dx
From the reduction formula
Sin^n(x) = -(1/n)cosx.sin^(n - 2)x + {(n - 1)/n}∫sin^(n - 2)x dx
sin^6x = -(1/6)cosx.sin^4x + 5/6∫sin^4x dx
sin^6x = -1/6cosx.sin^4x + 5/6{ -1/4cosx.sin^2x + 3/4∫sin^2x dx}
= -1/6cosx.sin^4x - 5/24cosx.sin^2x + 5/8∫sin^2x dx
= -1/6cosx.sin^4x - 5/24cosx.sin^2x + 5/8{ x/2 - 1/4sin2x}
∫sin^6(x) = -1/6cosx.sin^4x - 5/24cosx.sin^2x -5/32sin2x + 5/16x + c