# Math Help - Integration

1. ## Integration

$\displaystyle { \int \frac{1}{2-\sin x} dx }$

any hints how I can integrate it

2. Sub. $x = \frac{\pi}{2} - 2t$

we have

$\int \frac{dx}{2 - \sin(x) } = - 2 \int \frac{dt}{ 2 - \cos(2t) }$

$= -2 \int \frac{dt}{ 1 + (1 - \cos(2t) )}$

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

$= -2 \int \frac{dt}{ \cos^2(t) + 3\sin^2(t) }$

$= -2 \int \frac{ \sec^2(t) ~dt }{ 1 + 3 \tan^2(t) }$

$= - \frac{2}{\sqrt{3} } \tan^{-1}[ \sqrt{3} \tan(t) ] + C$

Then back substitution .

3. Thank you very much I appreciate that