Question

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

• A. $\frac{4}{\pi }$
• B. $\frac{8}{\pi }$
• C. $\frac{8}{{\pi }^2}$
• D. $0$

Answer 1

Answer: (D)

$0$

$I={\int }_0^{\pi }f\left(x\right)dx={\int }_0^{\pi }\frac{\sin 2x\sin (\frac{\pi }{2}.\cos x)}{2x-\pi }dx$
$\therefore$ $I={\int }_0^{\pi }\frac{\sin 2(\pi -x)\sin (\frac{\pi }{2}.\cos (\pi -x))}{2(\pi -x)-\pi }dx$
$={\int }_0^{\pi }\frac{(-1)\sin 2x.\sin \left(\frac{\pi }{2}\cos x\right)}{\pi -2x}dx=-I$
$\therefore$ $2I=0$ $\therefore$ $I=0$