Question

The value of ${\int }_0^{\pi }x{\sin }^{3}xdx$ is:

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

Answer 1

The correct option is B $\frac{2\pi }{3}$

Let $I={\int }_0^{\pi }x{\sin }^{3}xdx$ .....(i)

Also, $I={\int }_0^{\pi }(\pi -x){\sin }^{3}xdx$ ......(ii)

On adding equations (i) and (ii), we get

$2I=\pi {\int }_0^{\pi }{\sin }^{3}xdx$

$=\frac{\pi }{4}{\int }_0^{\pi }(3\sin x-\sin 3x)dx$

$=\frac{\pi }{4}{[-3\cos x+\frac{\cos 3x}{3}]}_0^{\pi }$

$=\frac{\pi }{4}[3-\frac{1}{3}+3-\frac{1}{3}]=\frac{4\pi }{3}$

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