Question

The value of ${\int }_0^{\pi /4}\frac{{\sin }^{\frac{1}{2}}x}{{\cos }^{\frac{5}{2}}x}dx$ is

• A. $0$
• B. $\frac{\pi }{4}$
• C. $1$
• D. $\frac{2}{3}$

Answer 1

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

${\int }_0^{\pi /4}\frac{{\sin }^{\frac{1}{2}}x}{{\cos }^{\frac{5}{2}}x}dx$

$={\int }_0^{\pi /4}\frac{{\tan }^{\frac{1}{2}}x}{{\cos }^{2}x}dx$

$={\int }_0^{\pi /4}{\sec }^{2}x{\tan }^{\frac{1}{2}}dx$

Now, let $z=\tan x$ $⟹dz={\sec }^{2}xdx$

And at $x=0,z=0$ and at $x=\frac{\pi }{4},z=1$

Now, integration becomes ${\int }_0^1\sqrt{z}dz$

$=\frac{2}{3}{\left[z^{\frac{3}{2}}\right]}_0^1$

$=\frac{2}{3}$