Question

Evaluate:

$\underset{0}{\overset{2a}{\int }}\frac{f\left(x\right)}{f\left(x\right)+f(2a-x)}dx$.

Answer 1

Let $I=$$\underset{0}{\overset{2a}{\int }}\frac{f\left(x\right)}{f\left(x\right)+f(2a-x)}dx$.......(1).

$I=$$\underset{0}{\overset{2a}{\int }}\frac{f(2a-x)}{f(2a-x)+f\left(x\right)}dx$ [ Using property of definite integral]......(2).

Now adding (1) and (2) we get,

$2I=$$\underset{0}{\overset{2a}{\int }}dx$

or, $2I=2a$

or, $I=a$.