Question

Solve $\int \frac{dx}{\sin x+\sqrt{3}\cos x}$

• A. $\frac{1}{2}\log \left|\tan (\frac{x}{2}-\frac{\pi }{6})\right|+c$
• B. $\frac{1}{2}\log \left|\tan (\frac{x}{4}-\frac{\pi }{6})\right|+c$
• C. $\frac{1}{2}\log \left|\tan (\frac{x}{2}+\frac{\pi }{6})\right|+c$
• D. $\frac{1}{2}\log \left|\tan (\frac{x}{4}+\frac{\pi }{3})\right|+c$
  where $c$ is an arbitrary constant

Answer 1

Answer: (C)

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

Let $I=\int \frac{dx}{\sin x+\sqrt{3}\cos x}$
$=\int \frac{dx}{2\sin (x+\frac{\pi }{3})}$
$=\frac{1}{2}\int \csc (x+\frac{\pi }{3})dx$
$=\frac{1}{2}\log \tan (\frac{x}{2}+\frac{\pi }{6})+c$