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The integral int(0)^(a) (g(x))/(f(x)+f(a...

The integral `int_(0)^(a) (g(x))/(f(x)+f(a-x))dx` vanishes, if

A

g(x) is odd

B

f(x) =f(a-x)

C

g(x)=-g(a-x)

D

f(a-x)g=g(x)

Text Solution

Verified by Experts

The correct Answer is:
C
Let `underset(0)overset(a)int (g(x))/(f(x)+f(a-x))dx` then,
`I=underset(0)overset(a)int (g(x))/(f(x)+f(a-x))[ :. underset(0)overset(a)intphi(x)dx=underset(0)overset(a)intphi(a-x)dx]`
` :. I+I=underset(0)overset(a)int (g(x)+g(a-x))/(f(x)+f(a-x))dx`
`rArr I(1)/(2)=underset(0)overset(a)int (g(x)+g(a-x))/(f(x)+f(a-x))dx`
Clearly, I vanishes if `g(a-x)=-g(x)`
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