Question

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{3}xdx$ is equal to

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $1$
• D. $\frac{4}{3}$

Answer 1

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

Let $I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{3}xdx$
$\{\because \sin 3x=3\sin x-4{\sin }^{3}x\}\Rightarrow I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{3\sin x-\sin 3x}{4}dx=\frac{3}{4}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin xdx-\frac{1}{4}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin 3xdx=\frac{3}{4}{[-\cos x]}_0^{\frac{\pi }{2}}-\frac{1}{4}{\left[\frac{-\cos 3x}{3}\right]}_0^{\frac{\pi }{2}}=\frac{3}{4}-\frac{1}{12}=\frac{2}{3}$