If ∫ cos x d x sin 3 x 1 + sin 6 x 2 3 = f x 1 + sin 6 x 1 λ + c , where c is a constant of integration, then λ f π 3 is equal to

Step-by-step solution

Let, s i n x = t c o s x d x = d t I = ∫ d t t 3 1 + t 6 2 3 = ∫ d t t 7 1 + 1 t 6 2 3 Put 1 + 1 t 6 = r 3 ⇒ d t t 7 = - 1 2 r 2 d r I = - 1 2 ∫ r 2 d r r 2 = - 1 2 r + c ⇒ I = - 1 2 sin 6 x + 1 sin 6 x 1 3 + c ⇒ I = - 1 2 sin 2 x 1 + sin 6 x 1 3 + c So, f x = - 1 2 c o s e c 2 x and λ = 3 λ f π 3 = - 2