Let f and g be continuous functions on 0 , a such that f x = f a - x and g x + g a - x = 4 , then ∫ 0 a f x g x d x is equal to

Question

Let f and g be continuous functions on 0 , a such that f x = f a - x and g x + g a - x = 4 , then ∫ 0 a f x g x d x is equal to, ∫ 0 a f x d x, - 3 ∫ 0 a f x d x, 4 ∫ 0 a f x d x, 2 ∫ 0 a f x d x

MARKS App · Accepted Answer

Let I = ∫ 0 a f x g x d x ⇒ I = ∫ 0 a f a - x g a - x d x (Using properties of definite integration) ⇒ I = ∫ 0 a f x 4 - g x d x ∵ f x = f a - x & g x + g a - x = 4 ⇒ I = 4 ∫ 0 a f x d x - ∫ 0 a f x g x d x ⇒ I = 4 ∫ 0 a f x d x - I ⇒ 2 I = 4 ∫ 0 a f x d x ⇒ I = 2 ∫ 0 a f x d x

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