If I x = ∫ e sin 2 x cos x sin 2 x - sin x d x and I 0 = 1 , then I π 3 is equal to

Step-by-step solution

Given, I x = ∫ e sin 2 x cos x   sin 2 x - sin x d x ⇒ I x = ∫ e sin 2 x cos x - 1 2   cos x sin 2 x   d x Now let, sin 2 x = t ⇒ sin 2 x   d x = d t I x = ∫ e t 1 - t - 1 2 1 - t d t Now comparing with, I x = ∫ e t f t + f ' t d t = e t f t + c Here, f t = 1 - t   &   f ' t = - 1 2 1 - t ⇒ I x = e t 1 - t + c = e sin 2 x · cos x + c Now using, I 0 = 1 ⇒ c = 0 Hence, the value of I π 3 = 1 2 e 3 4