Evaluate: `int1/(1-2sinx)\ dx`

Explanation

`sinx=frac{2tan x/2}{1+tan^2 x/2}`

`int frac{1}{1-2sinx}dx=int frac{1}{1-2 frac{2 tan x/2}{1+tan^2 x/2}}dx`

`=int frac{1+tan^2 x/2}{1(1+tan^2x/2)-2(2 tan x/2)}dx`

On replacing `(1 + tan^2x)/2` in numerator by `sec^2x/2`

`=int frac{sec^2 x/2}{1+tan^2 x/2-4tan x/2}dx`

`tan x/2=t=> sec^2 x/2dx=2dt`

`=2 int frac{1}{(1-2)^2-(sqrt 3)^2}dt`

`=1/(sqrt 3) log abs(frac{tan x/2-2- sqrt 3}{tan x/2-2+ sqrt 3})+C`