Question

The value of integral $\underset{0}{\overset{\pi }{\int }}xf\left(\sin x\right)dx$ is

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

Answer 1

Answer: (B)

Let $I=\underset{0}{\overset{\pi }{\int }}xf\left(\sin x\right)dx$

$\Rightarrow I=\underset{0}{\overset{\pi }{\int }}\left(\pi -x\right)f\left\{\sin \left(\pi -x\right)\right\}dx$

$=\pi \underset{0}{\overset{\pi }{\int }}f\left(\sin x\right)dx-x\underset{0}{\overset{\pi }{\int }}f(\sin x)dx$

$\Rightarrow I=\pi \underset{0}{\overset{\pi }{\int }}f(\sin x)dx-I$

$\Rightarrow 2I=\pi \underset{0}{\overset{\pi }{\int }}f\left(\sin x\right)dx$

$\Rightarrow 2I=2\pi \underset{0}{\overset{\pi /2}{\int }}f\left(\sin x\right)dx$

$\Rightarrow I=\pi \underset{0}{\overset{\pi /2}{\int }}f\left(\sin x\right)dx$