If $I=\int\limits_0^\pi x \sin x d x$, then
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) - $I=\pi$
$I=\int\limits_0^\pi x \sin x d x$ ...(1)
$I=\int\limits_0^\pi (\pi - x)\sin (\pi - x)dx$
$I=\int\limits_0^\pi (\pi - x)\sin xdx$ ...(2)
adding (1) and (2)
$⇒2I=\pi\int\limits_0^\pi \sin xdx$
$⇒2I=\pi[\cos x]_0^\pi$
$2I=\pi×2$
$I=\pi$