Question

Solve

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

Answer 1

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

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

$=\frac{1}{2}\int \frac{1}{\sin x\cos {60}^o+\sin {60}^o\cos x}dx$

$=\frac{1}{2}\int \frac{1}{\sin (x+\pi /3)}dx$

$\Rightarrow \frac{1}{2}\int \csc (x+\pi /3)dx$

$=\frac{1}{2}[-ln|\csc (x+\pi /3)+\cot (x+\pi /3\left)|\right]+C$