Question

$\int \frac{1}{\cos x+\sqrt{3}\sin x}dx$ is equal to

(a) $\log \tan \left(\frac{\mathrm{\pi }}{3}+\frac{x}{2}\right)+C$

(b) $\log \tan \left(\frac{x}{2}-\frac{\mathrm{\pi }}{3}\right)+C$

(c) $\frac{1}{2}\log \tan \left(\frac{x}{2}+\frac{\mathrm{\pi }}{3}\right)+C$

(d) none of these

Answer 1

(d) none of these

$$\begin{gathered} \int \frac{1}{\cos x+\sqrt{3}\sin x}dx \\ =\frac{1}{2}\int \frac{dx}{\cos x\times {\displaystyle \frac{1}{2}}+\sin x\times {\displaystyle \frac{\sqrt{3}}{2}}} \\ =\frac{1}{2}\int \frac{dx}{\cos x·\cos {\displaystyle \frac{\mathrm{\pi }}{3}}+\sin x·\sin {\displaystyle \frac{\mathrm{\pi }}{3}}} \\ =\frac{1}{2}\int \frac{dx}{\cos \left(x-{\displaystyle \frac{\mathrm{\pi }}{3}}\right)} \\ =\frac{1}{2}\int \sec \left(x-\frac{\mathrm{\pi }}{3}\right)dx \\ =\frac{1}{2}\ln \left|\tan \left\{\frac{\mathrm{\pi }}{4}+\frac{1}{2}\left(x-\frac{\mathrm{\pi }}{3}\right)\right\}\right|+C \\ =\frac{1}{2}\ln \left|\tan \left(\frac{\mathrm{\pi }}{4}+\frac{x}{2}-\frac{\mathrm{\pi }}{6}\right)\right|+C \\ =\frac{1}{2}\ln \left|\tan \left(\frac{x}{2}+\frac{\mathrm{\pi }}{12}\right)\right|+C \end{gathered}$$