Question

Integrate: ${\int }^{}\frac{1}{\sin x+\sqrt{3}\cos x}dx$

Answer 1

$\begin{array}{c}I={\int }^{}\frac{1}{\sin x+\sqrt{3}\cos x}dx\\ ={\int }^{}\frac{2}{2\left(\sin x+\sqrt{3}\cos x\right)}dx\\ 2I={\int }^{}\frac{1}{\frac{1}{2}\left(\sin x+\sqrt{3}\cos x\right)}dx\\ ={\int }^{}\frac{1}{\left(\frac{1}{2}\sin x+\sqrt{3}\cos x\right)}dx\\ ={\int }^{}\frac{1}{\sin 30\sin x+\cos 30\cos x}dx\\ ={\int }^{}\frac{1}{\cos \left(x-30\right)}dx\\ ={\int }^{}\sec \left(x-30\right)dx\\ I=\frac{1}{2}\log \left|\tan \left(x-30\right)+\sec \left(x-30\right)\right|+C\end{array}$