a-b-f-x-f-x-f-a-b-x-dx-

Question Number 259 by a@b.c last updated on 25/Jan/15

\int_{a}^{b} \frac{f (x)}{f (x) + f (a + b - x)} d x =

Answered by prakash jain last updated on 17/Dec/14

Substitue x = a + b - y \Rightarrow d x = - d y

The given integral I

I = \int_{b}^{a} \frac{f (a + b - y)}{f (a + b - y) + f (y)} (- d y) = \int_{b}^{a} \frac{f (a + b - y)}{f (a + b - y) + f (y)} d y

= \int_{a}^{b} \frac{f (a + b - y)}{f (a + b - y) + f (y)} d y = \int_{a}^{b} \frac{f (a + b - x)}{f (a + b - x) + f (x)} d x

2 I = \int_{a}^{b} \frac{f (a + b - x) + f (x)}{f (a + b - x) + f (x)} d x = \int_{a}^{b} 1 \cdot d x = b - a

I = \frac{b - a}{2}