Question

The value of the integral $\int\limits_0^\pi x \ell n \sin x d x$ is

• A. $-\frac{\pi^2}{2} \ell \ln 2$
• B. $\frac{\pi^2}{2} \ln 2$
• C. $\pi^2 \ell n 2$
• D. $-\pi^2 \ell n 2$

Answer 1

Answer: (A)

$I=\int\limits_0^\pi x \ell n \sin x d x$
$I=\int\limits_0^\pi(\pi-x) \ell n \sin (\pi-x) d x$
$=\int\limits_0^\pi(\pi-x) \ell n \sin x d x$
Add $2 I=\pi \int\limits_0^\pi \ell n \sin x d x$
$I=\frac{\pi}{2} \int\limits_0^\pi \ell n \sin x d x$
$=-\frac{\pi^2}{2} \ln 2$