Question

Solve the integral:

$\int \frac{1}{\sin x+\sqrt{3}\cos x}dx$

Answer 1

$l=\int \frac{1}{\sin x+\sqrt{3}\cos x}dx$

$l=\int \frac{2}{2(\sin x+\sqrt{3}\cos x)}dx$

$2l=\int \frac{2}{(\sin x+\sqrt{3}\cos x)}dx$

$2l=\int \frac{1}{\frac{1}{2}(\sin x+\sqrt{3}\cos x)}dx$

$2l=\int \frac{1}{(\frac{1}{2}\sin x+\frac{\sqrt{3}}{2}\cos x)}dx$

$2l=\int \frac{1}{\cos 30\sin x+\cos 30}dx$

$2l=\int \frac{1}{\cos \left(x-30\right)}dx$

$2l=\int \sec \left(x-30\right)dx$

$l=\frac{1}{2}\log \left|\tan \left(x-30\right)+\sec \left(x-30\right)\right|$