Question

Assertion :The value of ${\int }_0^{\pi }xf\left(\sin x\right)dx$ is $\frac{\pi }{2}{\int }_0^{\pi }f\left(\sin x\right)dx$ or $\pi {\int }_0^{\pi /2}f\left(\sin x\right)dx$ Reason:

${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$ and ${\int }_0^{2a}f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx$ If $f(2a-x)=f\left(x\right)$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $I={\int }_0^{\pi }xf\left(\sin x\right)dx$

$={\int }_0^{\pi }(\pi -x)f\left(\sin (\pi -x)\right)dx$
$(\because {\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx)$
$\Rightarrow$ $2I=\pi {\int }_0^{\pi }f\left(\sin x\right)dx$
$\Rightarrow$ $I=\frac{\pi }{2}{\int }_0^{\pi }f\left(\sin x\right)dx$
$=2.\frac{\pi }{2}{\int }_0^{\frac{\pi }{2}}f\left(\sin x\right)dx$ $(\because f\left(\sin (\pi -x)\right)=f\left(\sin x\right))$

$=\pi {\int }_0^{\frac{\pi }{2}}f\left(\sin x\right)dx$