Question

Evaluate: $\int \frac{dx}{{\cos }^{2}x{\left(1-\tan x\right)}^2}$

Answer 1

$I=\int \frac{1}{{\cos }^{2}x(1-\tan x{)}^2}dx$

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

Let $\tan x=t\Rightarrow {\sec }^{2}xdx=dt$

$I=\int \frac{dt}{(1-t{)}^2}$

$I=\frac{1}{(1-t)}+C$

$I=\frac{1}{(1-\tan x)}+C$