Question

Integrate the following functions.
$\int \frac{1}{co{s}^{2}x(1-tanx{)}^2}dx$.

Answer 1

$\int \frac{1}{co{s}^{2}x(1-tanx{)}^2}dx=\int \frac{se{c}^{2}x}{(1-tanx{)}^2}dx$
Let 1-tan x=t
$\Rightarrow -se{c}^{2}x=\frac{dt}{dx}\Rightarrow dx=\frac{dt}{-se{c}^{2}x}\therefore \int \frac{se{c}^{2}x}{(1-tanx{)}^2}dx=\int \frac{se{c}^{2}x}{t^2}\frac{dt}{(-se{c}^{2}x)}=-\int t^{-2}dt=-\left(\frac{-t^{-2+1}}{-2+1}\right)+C=\frac{1}{t}+C=\frac{1}{1-tanx}+C$