Question

The value of $\underset{0}{\overset{\pi /2}{\int }}\left[\frac{\left({\sin }^{1/3}x\right)}{({\sin }^{1/3}x+{\cos }^{1/3}x)}\right]dx$ is

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

Answer 1

The correct option is B $\frac{\pi }{4}$

Let $I=\underset{0}{\overset{\pi /2}{\int }}\left[\frac{\left({\sin }^{1/3}x\right)}{({\sin }^{1/3}x+{\cos }^{1/3}x)}\right]dx\cdots \left(i\right)$
Using property
$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f(a+b-x)dx$
We get,
$I=\underset{0}{\overset{\pi /2}{\int }}\left[\frac{\left({\cos }^{1/3}x\right)}{({\sin }^{1/3}x+{\cos }^{1/3}x)}\right]dx\cdots \left(ii\right)$

On adding $\left(i\right)$ and $\left(ii\right),$ we get
$\Rightarrow 2I=\frac{\pi }{2}$
$\therefore I=\frac{\pi }{4}$