Question

$\int _ { 0 } ^ { \pi } x \sin ^ { 3 } x d x =$

• A. $\frac { 4 \pi } { 3 }$
• B. $\frac { 2 \pi } { 3 }$
• C. 0
• D. None of these

Answer 1

Answer: (B)

$\frac { 2 \pi } { 3 }$

Answer 2

Let $I = \int _ { 0 } ^ { \pi } x \sin ^ { 3 } x d x$ …..(i)

Also $I = \int _ { 0 } ^ { \pi } ( \pi - x ) \sin ^ { 3 } x d x$ …..(ii)

Adding (i) and (ii), we get

$2 I = \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } x d x = \frac { \pi } { 4 } \int _ { 0 } ^ { \pi } \{ 3 \sin x - \sin 3 x \} d x$

$= \frac { \pi } { 4 } \left[ - 3 \cos x + \frac { \cos 3 x } { 3 } \right] _ { 0 } ^ { \pi } = \frac { \pi } { 4 } \left[ 3 - \frac { 1 } { 3 } + 3 - \frac { 1 } { 3 } \right] = \frac { 4 \pi } { 3 }$

Hence, $I = \frac { 2 \pi } { 3 }$ .