Question

Evaluate: $\int \frac{dx}{\sin x-\sqrt{3}\cos x}$

Answer 1

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

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

$=\frac{1}{2}\int \frac{dx}{\cos \frac{\pi }{3}\sin x-\sin \frac{\pi }{3}\cos x}$

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

$=\frac{1}{2}\int \csc (\frac{\pi }{3}-x)dx$

$=\frac{1}{2}\ln |\csc (\frac{\pi }{3}-x)-\cot (\frac{\pi }{3}-x)|+c$