Question

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

Answer 1

We have,

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

On divide and multiply denominator

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

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

$=\frac{1}{2}\int \frac{dx}{(\sin x\cos \frac{\pi }{6}+\cos x\sin \frac{\pi }{6})}$

$=\frac{1}{2}\int \frac{dx}{\sin (x+\frac{\pi }{6})}$

$=\frac{1}{2}\int \csc (x+\frac{\pi }{6})dx$

$=-\frac{1}{2}\ln [\csc (x+\frac{\pi }{6})+\cot (x+\frac{\pi }{6})]+C$

Hence, this is the answer.