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MHT CET20242 May 2024Morning 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. A4 ₀^a f (x) d x
  2. B₀^a f (x) d x
  3. C2 ₀^ a f (x) d x
  4. D-3 ₀^a f (x) d x

Correct answer

C. 2 ₀^ a f (x) d x

Step-by-step solution

Given, aligned & f (x)= f ( a - x ) & g (x)+ g ( a -x)=4 & Let I = ₀^ a f (x) g (x) d x & = ₀^a f(a-x) g(a-x) d x & = ₀^ a f (x) [(4- g (x)] d x & =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 aligned

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