Q115 / 225 · JEE Main 2004 · Mathematics

If $\int_0^\pi x f(\sin x) d x=A \int_0^{\pi / 2} f(\sin x) d x$, then $A$ is

  • A
    0
  • Correct answer
    B
    $\pi$
  • C
    $\frac{\pi}{4}$
  • D
    $2 \pi$

Solution

Let $I=\int_0^\pi x f(\sin x) d x=\int_0^\pi(\pi-x) f(\sin x) d x=\pi \int_0^\pi f(\sin x) d x-I \quad(\sin c e f(2 a-x)=f(x))$ $\Rightarrow I=\pi \int_0^{\pi / 2} f(\sin x) d x \Rightarrow A=\pi$

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

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