Question

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

Answer 1

$\int \frac{dx}{sinx+\sqrt{3}cosx}$

$=\frac{1}{2}\int \frac{1}{(\frac{1}{2}sinx+\frac{\sqrt{3}}{2}cosx)}dx$

$=\frac{1}{2}\int \frac{1}{(\frac{\sqrt{3}}{2}cosx+\frac{1}{2}sinx)}dx$

$=\frac{1}{2}\int \frac{1}{\sin \frac{\Pi }{3}cosx+sinx\cos \frac{\Pi }{3}}dx$ [$\sin \frac{\Pi }{3}=\frac{\sqrt{3}}{2}$]

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

$=\frac{1}{2}\int cosec(x+\frac{\Pi }{3})dx$

$=\frac{1}{2}log\left|\tan \left(\frac{x+\Pi /3}{2}\right)\right|+C$