Question

Evaluate: $\int \frac{\sqrt{\sin x}}{{\cos x}^{5/2}}dx$

• A. $\frac{2}{3}{\tan }^{3/2}x+c$
• B. $\frac{2}{3}{\tan }^{2/3}x+c$
• C. $\frac{3}{2}{\tan }^{3/2}x+c$
• D. $\frac{3}{2}{\tan }^{2/3}x+c$

Answer 1

Answer: (A)

$\frac{2}{3}{\tan }^{3/2}x+c$

Let $I=\int \frac{\sqrt{\sin x}}{{\cos }^{5/2}x}dx=\int \sqrt{\tan x}{\sec }^{2}xdx$
Put $\tan x=t\Rightarrow {\sec }^{2}xdx=dt$
$I=\int \sqrt{t}dt=\frac{2t^{3/2}}{3}=\frac{2}{3}{\tan }^{3/2}x$