Question

The value of integral $\underset{0}{\overset{\pi /2}{\int }}\frac{{\sin }^{5/2}x}{{\sin }^{5/2}x+{\cos }^{5/2}x}dx$ is

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

Answer 1

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

Let $I=\underset{0}{\overset{\pi /2}{\int }}\frac{{\sin }^{5/2}x}{{\sin }^{5/2}x+{\cos }^{5/2}x}dx\cdots \left(i\right)$

$\Rightarrow$ Using property, we get
$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f(a+b-x)dx$
$I=\underset{0}{\overset{\pi /2}{\int }}\frac{{\cos }^{5/2}x}{{\sin }^{5/2}x+{\cos }^{5/2}x}dx\cdots \left(ii\right)$

On adding $\left(i\right)$ and $\left(ii\right),$ we get
$2I=\underset{0}{\overset{\pi /2}{\int }}1dx=\frac{\pi }{2}$
$\Rightarrow I=\frac{\pi }{4}$