Question

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

Answer 1

$I=\int \frac{1}{co{s}^{2}x(1-tanx{)}^2}dx$

$I=\int \frac{se{c}^{2}x}{(1-tanx{)}^2}dx$

$1-tanx=t$

$-se{c}^{2}xdx=dt$

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

$=-\frac{t^{-2+1}}{(-2+1)}+c$

$I=+t^{-1}+c$

$I=\frac{1}{t}+c$

$I=\frac{1}{(1-tanx)}+c$

$\therefore \int \frac{1}{co{s}^{2}x(1-tanx{)}^2}dx=\frac{1}{(1-tanx)}+c$