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  4. The value of integral ∫02a(f(x)/f(x)+f(2a-x))dx, where f(x) is a continuous function, is

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Q. The value of integral $ \int_{0}^{2a}{\frac{f(x)}{f(x)+f(2a-x)}}dx, $ where $ f(x) $ is a continuous function, is

Rajasthan PETRajasthan PET 2009

Solution:

Let $ I=\int_{0}^{2a}{\frac{f(x)}{f(x)+f(2a-x)}}dx $ ...(i)
$ \Rightarrow $ $ I=\int_{0}^{2a}{\frac{f(2a-x)}{f(x)+f(2a-x)}}dx $ ..(ii)
On adding Eqs. (i) and (ii), we get
$ 2I=\int_{0}^{2a}{1\,}dx\Rightarrow 2I[x]_{0}^{2a} $
$ \Rightarrow $ $ 2I=2a $
$ \Rightarrow $ $ I=a $