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JEE Main201912 Jan 2019Morning ShiftMathematicsDefinite IntegrationActual

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

Options

  1. A∫ 0 a f x d x
  2. B- 3 ∫ 0 a f x d x
  3. C4 ∫ 0 a f x d x
  4. D2 ∫ 0 a f x d x

Correct answer

D. 2 ∫ 0 a f x d x

Step-by-step solution

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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