If the value of the integral ∫ - π 2 π 2 x 2 cos x 1 + π x + 1 + sin 2 x 1 + e sin x 2023 d x = π 4 π + a - 2 , then the value of a is

Step-by-step solution

Given, ∫ - π 2 π 2 x 2 cos x 1 + π x + 1 + sin 2 x 1 + e sin 2023 x d x = π 4 π + α - 4 Now let, I 1 = ∫ - π 2 π 2 x 2 cos x 1 + π x d x . . . . . . . i Now using the property ∫ a b f x d x = ∫ a b f a + b - x d x we get, ⇒ I 1 = ∫ - π 2 π 2 x 2 cos x 1 + π - x d x . . . . . . . . i i Now adding above equations we get, ⇒ 2 I 1 = 2 ∫ 0 π 2 x 2 cos x d x as ∫ - a a f x d x = 2 ∫ 0 a f x d x , if f x is even ⇒ I 1 = x 2 sin x 0 π 2 - ∫ 0 π 2 2 x · sin x d x ⇒ I 1 = π 2 4 - - 2 x cos x 0 π 2 + ∫ 0 π 2 2 cos x d x ⇒ I 1