Question

The value of integral $\int\limits^{\pi}_{0}xf\left(\sin\,x\right)dx$ is

• A. $0$
• B. $\pi\,\int\limits^{\pi/2}_{0}f\left(\sin\,x\right)dx$
• C. $\frac{\pi}{4}\,\int\limits^{\pi}_{0}f\left(\sin\,x\right)dx$
• D. None of these

Answer 1

Answer: (B)

Let $I=\int\limits^{\pi}_{0}xf\left(\sin\,x\right)dx$

$\Rightarrow I=\int\limits^{\pi}_{0}\left(\pi-x\right)\,f\,\left\{\sin\left(\pi-x\right)\right\}dx$

$=\pi\,\int\limits^{\pi}_{0}\,f\left(\sin\,x\right)dx-x\, \int\limits^{\pi}_{0} \,f(\sin\,x)dx$

$\Rightarrow I=\pi \int\limits^{\pi}_{0} \,f(\sin\,x)dx-I$

$\Rightarrow 2I=\pi\,\int\limits^{\pi}_{0}\,f\left(\sin\,x\right)dx$

$\Rightarrow 2I=2\pi\,\int\limits^{\pi/2}_{0}\,f\left(\sin\,x\right)dx$

$\Rightarrow I=\pi\,\int\limits^{\pi/2}_{0}\,f\left(\sin\,x\right)dx$