integrate (sin^6 x) dx

Question

Integration

Answer 1

∫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