Question

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

• A. $\frac{1}{2}\log \left|\frac{x}{2}+\frac{\pi }{12}\right|+c$
• B. $\frac{1}{2}\log \left|\frac{x}{2}-\frac{\pi }{12}\right|+c$
• C. $\frac{1}{2}\log \left|\tan \frac{x}{2}+\frac{\pi }{2}\right|+c$
• D. $\frac{1}{2}\log \left|\tan \frac{\pi }{12}+\frac{x}{2}\right|+c$

Answer 1

The correct option is D

$\frac{1}{2}\log \left|\tan \frac{\pi }{12}+\frac{x}{2}\right|+c$

Explanation for the correct option.

Finding the integral.

Given, $\int \frac{dx}{(\cos x+\sqrt{3}\sin x)}$

Solving the integral,

$\begin{array}{l}\begin{array}{cc}\int \frac{dx}{\cos x+\sqrt{3}\sin x}& =\int \frac{dx}{2\left(\frac{1}{2}\cos x+\frac{\sqrt{3}}{2}\sin x\right)}\\ & =\frac{1}{2}\int \frac{dx}{\left(\sin \frac{\pi }{6}\cos x+\cos \frac{\pi }{6}\sin x\right)}\\ & =\frac{1}{2}\int \frac{dx}{\sin \left(\frac{\pi }{6}+x\right)}\\ & =\frac{1}{2}\int \mathrm{cosec}\left(\frac{\pi }{6}+x\right)dx\\ & =\frac{1}{2}\log \left|\mathrm{cosec}\left(\frac{\pi }{6}+x\right)-\cot \left(\frac{\pi }{6}+x\right)\right|+c\\ & =\frac{1}{2}\log \left|\frac{(1-\cos \left(\frac{\pi }{6}+x\right)}{\sin \left(\frac{\pi }{6}+x\right)}\right|+c\\ & =\frac{1}{2}\log \left|\frac{2{\sin }^2\left(\frac{\pi }{12}+\frac{x}{2}\right)}{2\sin \left(\frac{\pi }{12}+\frac{x}{2}\right)\cos \left(\frac{\pi }{12}+\frac{x}{2}\right)}\right|+c\\ =& \frac{1}{2}\log \left|t\mathrm{an}\left(\frac{\pi }{12}+\frac{x}{2}\right)\right|+c\end{array}\end{array}$

Hence, the correct answer is Option (D)