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Question: The integral \(\int_{0}^{a}\frac{g(x)}{f(x) + f(a–x)}\) vanishes if -...

The integral ∫0ag(x)f(x)+f(a–x)\int_{0}^{a}\frac{g(x)}{f(x) + f(a–x)} vanishes if -

A

g(x) is odd

B

f(x) = f (a – x)

C

g(x) = – g(a – x)

D

f(a – x) = g(x)

Answer

g(x) = – g(a – x)

Explanation

Solution

I = ∫0ag(x)f(x)+f(a–x)\int_{0}^{a}\frac{g(x)}{f(x) + f(a–x)} then I = ∫0ag(a–x)f(x)+f(a–x)\int_{0}^{a}\frac{g(a–x)}{f(x) + f(a–x)}

Ž 2I = ∫0ag(x)+g(a–x)f(x)+f(a–x)\int_{0}^{a}\frac{g(x) + g(a–x)}{f(x) + f(a–x)}

Ž I = 12∫0ag(x)+g(a–x)f(x)+f(a–x)\frac{1}{2}\int_{0}^{a}\frac{g(x) + g(a–x)}{f(x) + f(a–x)}

= 0 (vanishes) if –g(x) = g(a – x)