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?

Question

MARKS Content Team · Accepted Answer

Correct answer: (B) ∫_0^a f(x) dx. 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))

If $f(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ are continuous functions satisfying $f(\mathrm{x})=f(\mathrm{a}-\mathrm{x})$ and $\mathrm{g}(\mathrm{x})+\mathrm{g}(\mathrm{a}-\mathrm{x})=2$, then what is $\int_0^{\mathrm{a}} f(\mathrm{x}) \mathrm{g}(\mathrm{x}) \mathrm{dx}$ equal to?

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