Q56 / 160 · WBJEE 2009 · Mathematics

If $f(x)=f(a-x)$ then $\int_0^a x f(x) d x$ is equal to

  • A
    $\int_0^a f(x) d x$
  • B
    $\frac{a^2}{2} \int_0^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$
  • Correct answer
    C
    $\frac{a}{2} \int_0^a f(x) d x$
  • D
    $-\frac{a}{2} \int_0^{\mathrm{a}} f(x) d x$

Solution

Hints : $f(x)=f(a-x), I=\int_0^a x f(x) d x=\int_0^a(a-x) f(a-x) d x$ $$ \begin{aligned} & =\int_0^a(a-x) f(x) d x=a \int_0^a f(x) d x-I \\ & \therefore 2 I=a \int_0^a f(x) d x \Rightarrow I=\frac{a}{2} \int_0^a f(x) d x \end{aligned} $$

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

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