Question

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

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{dx}{\sin x-\sqrt{3}\cos x} \\ \mathrm{Putting}\sin x=\frac{2\tan {\displaystyle \frac{x}{2}}}{1+{\tan }^2{\displaystyle \frac{x}{2}}}\mathrm{and}\cos x=\frac{1-{\tan }^2{\displaystyle \frac{x}{2}}}{1+{\tan }^2{\displaystyle \frac{x}{2}}} \\ \Rightarrow I=\int \frac{1}{{\displaystyle \frac{2\tan {\displaystyle \frac{x}{2}}}{1+{\tan }^2{\displaystyle \frac{x}{2}}}-\sqrt{3}\frac{1-{\tan }^2{\displaystyle \frac{x}{2}}}{1+{\tan }^2{\displaystyle \frac{x}{2}}}}}dx \\ =\int \frac{1+{\tan }^2{\displaystyle \frac{x}{2}}}{2\tan {\displaystyle \frac{x}{2}}-\sqrt{3}+\sqrt{3}{\tan }^2{\displaystyle \frac{x}{2}}}dx \\ =\int \frac{{\sec }^2{\displaystyle \frac{x}{2}}}{\sqrt{3}{\tan }^2{\displaystyle \frac{x}{2}}+2\tan {\displaystyle \frac{x}{2}}-\sqrt{3}}dx \end{gathered}$$

$$\begin{gathered} \mathrm{Let}\tan \frac{x}{2}=t \\ \Rightarrow \frac{1}{2}{\sec }^2\frac{x}{2}dx=dt \\ \Rightarrow {\sec }^2\frac{x}{2}dx=2dt \\ \therefore I=2\int \frac{dt}{\sqrt{3}{t}^2+2t-\sqrt{3}} \\ =\frac{2}{\sqrt{3}}\int \frac{dt}{t^2+{\displaystyle \frac{2}{\sqrt{3}}t}-1} \\ =2\int \frac{dt}{t^2+{\displaystyle \frac{2}{\sqrt{3}}}t+{\left({\displaystyle \frac{{\displaystyle 1}}{\sqrt{3}}}\right)}^2-1-{\left({\displaystyle \frac{1}{\sqrt{3}}}\right)}^2{\displaystyle }} \\ =2\int \frac{dt}{{\left(t+{\displaystyle \frac{1}{\sqrt{3}}}\right)}^2-{\left({\displaystyle \frac{{\displaystyle 2}}{\sqrt{3}}}\right)}^2} \\ =\frac{2}{2\times {\displaystyle \frac{2}{\sqrt{3}}}}\log \left|\frac{t+{\displaystyle \frac{1}{\sqrt{3}}}-{\displaystyle \frac{2}{\sqrt{3}}}}{t+{\displaystyle \frac{1}{\sqrt{3}}}+{\displaystyle \frac{{\displaystyle 2}}{\sqrt{3}}}}\right|+C \end{gathered}$$
$$\begin{gathered} =\frac{\sqrt{3}}{2}\log \left|\frac{\tan {\displaystyle \frac{x}{2}}-{\displaystyle \frac{1}{\sqrt{3}}}}{\tan {\displaystyle \frac{x}{2}}+{\displaystyle \frac{3}{\sqrt{3}}}}\right|+C \\ =\frac{\sqrt{3}}{2}\log \left|\frac{\sqrt{3}\tan {\displaystyle \frac{x}{2}}-{\displaystyle 1}}{\sqrt{3}\tan {\displaystyle \frac{x}{2}}+3}\right|+C \end{gathered}$$