Question

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

Answer 1

We have,

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

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

$I=\frac{1}{2}\int \left(\frac{1}{\sin \frac{\pi }{3}\sin x+\cos \frac{\pi }{3}\cos x}\right)dx$

$I=\frac{1}{2}\int \left(\frac{1}{\cos (x-\frac{\pi }{3})}\right)dx$

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

We know that

$\int \sec xdx=\log |\sec x+\tan x|+C$

Therefore,

$I=\frac{1}{2}\left[\log |\sec (x-\frac{\pi }{3})+\tan (x-\frac{\pi }{3})|\right]+C$

Hence, the value of integral is $\frac{1}{2}\left[\log |\sec (x-\frac{\pi }{3})+\tan (x-\frac{\pi }{3})|\right]+C$.