Let f x be a function satisfying f x + f π - x = π 2 , ∀ x ∈ ℝ . Then ∫ 0 π f x sin x d x is equal to

Step-by-step solution

Let I = ∫ 0 π f x sin x d x       . . . 1 Now using the property ∫ a b f x d x = ∫ a b f a + b - x d x we get, ⇒ I = ∫ 0 π f π - x sin π - x d x ⇒ I = ∫ 0 π f π - x sinx d x       . . . 2 Adding 1   &   2 , we get 2 I = ∫ 0 π f x + f π - x sinx d x ⇒ 2 I = π 2 ∫ 0 π sinx d x as given f x + f π - x = π 2 ⇒ 2 I = π 2 - cosx 0 π ⇒ 2 I