Question

Find $\int \frac{1}{\sin x{\cos }^{3}x}dx$

Answer 1

$\int \frac{1}{\sin x{\cos }^{3}x}dx$

$=\int \frac{1/{\cos }^{4}x}{\frac{\sin x{\cos }^{3}x}{{\cos }^{4}x}}dx$

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

Now $\tan x=t\to {\sec }^{3}xdx=dt$

$\Rightarrow \int \frac{1+t^2}{t}=dt$

$\Rightarrow \int \frac{1}{t}i+dt$

$\Rightarrow logt+\frac{t^2}{2}+c$

$\Rightarrow log\tan t+\frac{{\tan }^{2}x}{2}+c$.