State whether the given statement is true or false
the value of the integral $\int_{0}^{2 a} \frac{f (x)}{f (x) + f (2 a - x)} d x$ is equal to a.

True

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False

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Solution

The correct option is A True
Let $I = \int_{0}^{2 a} \frac{f (x)}{f (x + f (2 a - x))} d x$ ...(1)
Using $\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$
$I = \int_{0}^{2 a} \frac{f (2 a - x)}{f (2 a - x) + f (x)} d x$ ...(2)
Adding (1) and (2)
$2 I = \int_{0}^{2 a} d x = 2 a$

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