Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
$\int_0^\pi x \sin ^3 x d x=$
A
$\frac{4 \pi}{3}$
✓
$\frac{2 \pi}{3}$
C
$0$
D
$\frac{\pi}{4}$
✓
Answer
Correct option: B.
$\frac{2 \pi}{3}$
(B) Let $I =\int_0^\pi x \sin ^3 x d x$ ...(i) $=\int_0^\pi(\pi-x) \sin ^3 x d x$ ...(ii) $\ldots\left[\because \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$ Adding (i) and (ii), we get $2 I=\pi \int_0^\pi \sin ^3 x d x=\frac{\pi}{4} \int_0^\pi(3 \sin x-\sin 3 x) d x$ $\begin{array}{l}=\frac{\pi}{4}\left[-3 \cos x+\frac{\cos 3 x}{3}\right]_0^\pi \\ =\frac{\pi}{4}\left[3-\frac{1}{3}+3-\frac{1}{3}\right]=\frac{4 \pi}{3}\end{array}$ $\therefore \quad I=\frac{2 \pi}{3}$
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