Q117 / 160 · TS EAMCET 2022 (20 Jul Shift 2) · Mathematics

If $\int_{-a}^a f(x) d x=\int_0^a f(x) d x+\int_0^a g(x) d x$ then $g(x)=$

  • A
    $-f(x)$
  • B
    $f(x)$
  • Correct answer
    C
    $f(-x)$
  • D
    $f(x)+f(-x$

Solution

Since, $\int_{-a}^a f(x) d x=2 \int_0^a f(x) d x$ $$ \begin{aligned} & =\int_0^a f(x) d x=f \int_0^a f(x) d x \\ & =\int_0^a f(x) d x=f \int_0^a f(-x) d x \end{aligned} $$ Here $f(-x)=f(x)$ is an even function $$ \Rightarrow \mathrm{g}(\mathrm{x})=\mathrm{f}(-\mathrm{x}) $$

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

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