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NDA2017MathematicsDefinite IntegrationActual

If f( x ) and g ( x ) are continuous functions satisfying f( x )=f( a - x ) and g ( x )+ g ( a - x )=2 , then what is ₀^ a f( x ) g ( x ) dx equal to?

Options

  1. A₀^a g(x) d x
  2. B₀^ a f( x ) dx
  3. C2 ₀^ a f( x ) dx
  4. D0

Correct answer

B. ₀^ a f( x ) dx

Step-by-step solution

aligned & I= ₀^a f(x) g(x) d x & I= ₀^a f(a-x) g(a-x) d x & I= ₀^a f(x) [2-g(x)] d x & f(a-x)=f(x) and g(a-x)=2 g(x) & I= ₀^a 2 f(x) d x- ₀^a f(x) g(x) d x & I=2 ₀^a f(x) -I & 2 I=2 ₀^a f(x) d x 2 & I= ₀^a f(x) d x & a a f(x)= ₀^a f(a-x) d x & ( Given f(x)=f(x)) aligned

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