Question

$\int \frac{dx}{\sqrt{{\sin }^{3}x{\cos }^{5}x}}$

Answer 1

$\int \frac{1}{\sqrt{{\sin }^{3}x{\cos }^{5}x}}dx$
$=\int \frac{1}{\sqrt{{\cos }^{\frac{5}{2}}x{\sin }^{\frac{5}{2}}x}}dx$
$\because \sin x=\frac{\tan x}{\sec x}\cos x=\frac{1}{\sec x}$
${\sec }^{2}x={\tan }^{2}x+1$
$\Rightarrow \int {\sec }^{2}x\frac{{\tan }^2\left(x\right)+1}{{\tan }^{\frac{3}{2}}x}dx$
Substitute $u=\tan x\Rightarrow dx=\frac{1}{{\sec }^{2}x}du$
$\Rightarrow \int \frac{u^2+1}{u^{\frac{3}{2}}}du$
$\Rightarrow \int \sqrt{u}+\frac{1}{u^{\frac{3}{2}}}du$
$=\frac{2u^{\frac{3}{2}}}{3}-\frac{2}{\sqrt{u}}$
$=\frac{2{\tan }^{\frac{3}{2}}x}{3}-\frac{2}{\sqrt{\tan x}}=\frac{2({\tan }^{2}x-3)}{3\sqrt{\tan x}}+c$