Question

Evaluate: $\int \frac{dx}{\sin x{\cos }^{3}x}$.

Answer 1

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

$={\int }^{}\frac{{\sec }^{4}x}{\frac{\sin x}{\cos x}}dx$

putting $\tan x=t$

$\Rightarrow {\sec }^{2}xdx=dt$

$={\int }^{}\frac{\left(1+{\tan }^{2}x\right){\sec }^{2}x}{\tan x}dx$

$={\int }^{}\frac{\left(1+t^2\right)}{t}dt$

$={\int }^{}\left(\frac{1}{t}+t\right)dt$

$=\log \left|t\right|+\frac{t^2}{2}+c$

$=\log \left|\tan x\right|+\frac{{\tan }^{2}x}{2}+c$