int_0^{2a} frac{f(x)}{f(x) + f(2a - x)} , dx | vedclass

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(A) Let $I = \int_0^{2a} \frac{f(x)}{f(x) + f(2a - x)} \, dx$ ..... $(i)$Using the property $\int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx$,we have:$I

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$a$

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(A) Let $I = \int_0^{2a} \frac{f(x)}{f(x) + f(2a - x)} \, dx$ ..... $(i)$Using the property $\int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx$,we have:$I = \int_0^{2a} \frac{f(2a - x)}{f(2a - x) + f(2a - (2a - x))} \, dx$$I = \int_0^{2a} \frac{f(2a - x)}{f(2a - x) + f(x)} \, dx$ ..... $(ii)$Adding $(i)$ and $(ii)$:$2I = \int_0^{2a} \left( \frac{f(x)}{f(x) + f(2a - x)} + \frac{f(2a - x)}{f(2a - x) + f(x)} \right) \, dx$$2I = \int_0^{2a} \frac{f(x) + f(2a - x)}{f(x) + f(2a - x)} \, dx$$2I = \int_0^{2a} 1 \, dx$$2I = [x]_0^{2a} = 2a$$I = a$

$\int_0^{2a} \frac{f(x)}{f(x) + f(2a - x)} \, dx = $

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