Question

The value of $\int \frac{dx}{\sqrt{3}\sin x+\cos x}$ equals?

Answer 1

We are given $I=\int \frac{dx}{\sqrt{3}\sin x+\cos x}$

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

(Multiple & Divide $1/2$ in Numerator & Denominator)

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

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

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

We know that $\int \csc xdx=\log \left|\tan \frac{x}{2}\right|+C$

$I=\frac{1}{2}\log \left|\tan \frac{(x+\frac{\pi }{6})}{2}\right|+C$

$=\frac{1}{2}\log \left|\tan (\frac{x}{2}+\frac{\pi }{12})\right|+C$

So,

$\int \frac{dx}{\sqrt{3}\sin x+\cos x}=\frac{1}{2}\log \left|\tan (\frac{x}{2}+\frac{\pi }{12})\right|+C$