The value of the integral 48 π 4 ∫ 0 π 3 π x 2 2 - x 3 sin x 1 + cos 2 x d x is equal to ______.

Step-by-step solution

Let I = 48 π 4 ∫ 0 π 3 π 2 x 2 - x 3 sin x 1 + cos 2 x d x               … 1 Using ∫ a b f x d x = ∫ a b f ( a + b - x ) d x We get I = 48 π 4 ∫ 0 π 3 π 2 π - x 2 - π - x 3 sin 0 + π - x 1 + cos 2 0 + π - x d x I = 48 π 4 ∫ 0 π 3 π 3 2 - 3 π 2 x + 3 π x 2 2 - π 3 + x 3 + 3 π 2 x - 3 π x 2 sin x 1 + cos 2 x d x ..............equation 2 Now adding equation (1