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

Explanation

Let `=int frac{1}{1-2 sin x} d x`

Putting `sin x=frac{2 tan frac{x}{2}}{1+tan ^{2} frac{x}{2}}`
`
therefore I=int frac{1}{1-2(frac{2 tan frac{x}{2}}{1+tan ^{2} frac{7}{2}})} d x`
`
=int frac{(1+tan ^{2} frac{z}{2})}{1+tan ^{2} frac{z}{2}-4 tan frac{x}{2}} d x`
`
=int frac{sec ^{2} frac{z}{2}}{1+tan ^{2} frac{x}{2}-4 tan frac{z}{2}} d x`

Putting `tan frac{x}{2}=t`
`
Rightarrow frac{1}{2} sec ^{2}(frac{x}{2}) d x=d t`
`
Rightarrow sec ^{2}(frac{x}{2}) cdot d x=2 d t`
`
...