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…

Let f and g be continuous functions on 0,a such that fx=fa-x and gx+ga-x=4, then ∫0afxgxdx is equal to
  1. ∫0afxdx
  2. -3∫0afxdx
  3. 4∫0afxdx
  4. 2∫0afxdx

Solution

Let I=∫0afxgxdx

⇒I=∫0afa-xga-xdx    (Using properties of definite integration)

⇒I=∫0afx4-gxdx  ∵  fx=fa-x & gx+ga-x=4

⇒I=4∫0afxdx-∫0afxgxdx

⇒I=4∫0afxdx-I

⇒2I=4∫0afxdx

⇒I=2∫0afxdx

Asked in: JEE Main 2019 (12 Jan Shift 1)

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