The value of the integral ∫ 0 π 2 60 sin 6 x sin x d x is equal to

Step-by-step solution

Let I = ∫ 0 π 2 60 sin 6 x sin x d x ⇒ I = ∫ 0 π 2 60 2 sin 3 x cos 3 x sin x d x ⇒ I = 60 ∫ 0 π 2 2 3 sin x - 4 sin 3 x 4 cos 3 x - 3 cos x sin x d x ⇒ I = 120 ∫ 0 π 2 3 - 4 sin 2 x 4 cos 2 x - 3 cos x d x ⇒ I = 120 ∫ 0 π 2 3 - 4 sin 2 x 1 - 4 sin 2 x cos x d x Now let sin x = t ⇒ cos x d x = d t and limit changes to 0   to  1 , So, I = 120 ∫ 0 1 3 - 4 t 2 1 - 4 t 2 d t ⇒ I = 120 ∫ 0 1 3 - 16 t 2 + 16 t