Q228 / 270 · NDA 2021 (Phase 1) · Mathematics

What is $\int_0^a \frac{f(a-x)}{f(x)+f(a-x)} d x$ equal to?

  • A
    $a$
  • B
    $2 a$
  • C
    $0$
  • Correct answer
    D
    $\frac{a}{2}$

Solution

$\begin{aligned} I & =\int_0^a \frac{f(a-x)}{f(x)+f(a-x)} \cdot d x \\ & =\int_0^a \frac{f(a+0-a+x)}{f(a+0-x)+f(a+0-a+x)} \cdot d x\end{aligned}$ $\begin{aligned} & =\int_0^a \frac{f(x)}{f(a-x)+f(x)} \cdot d x \\ & 2 I=\int_0^a \frac{f(a-x)+f(x)}{f(a-x)+f(x)} \cdot d x=[x]_0^a\end{aligned}$ $\therefore I=\frac{a}{2}$

Hence, the correct answer is option: 'D'.

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