The value of ∫ - π 2 π 2 x 2 cos ⁡ x 1 + e x d x is equal to

Step-by-step solution

Let I = ∫ −   x 2 x 2 x 2 cos x 1 + e x d x Replacing x by - x I = ∫ − π 2 π 2 ( x 2 cos x 1 + e − x ) d x ⇒ 2 I = ∫ − π 2 π 2 ( 1 1 + e x + 1 1 + e − x ) x 2 cos x   d x ⇒ 2 I = ∫ − π 2 π 2 x 2 cos x   d x , (even function) I =   ∫ 0 π 2 x 2 cos ⁡ x   d x = x 2 sin ⁡ x 0 π 2 - 2 ∫ 0 π 2 x . sin ⁡ x   d x (Using integration by parts, taki