Question

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

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

Answer 1

The correct option is A $\frac{1}{2}\log \tan (\frac{x}{2}+\frac{\pi }{12})+c$

$\int \frac{dx}{\cos x+\sqrt{3}\sin x}=\frac{1}{2}\int \sec (x-\frac{\pi }{3})dx$
$=\frac{1}{2}\log \left(\tan (\frac{x}{2}-\frac{\pi }{6}+\frac{\pi }{4})\right)+c=\frac{1}{2}\log \left(\tan (\frac{x}{2}+\frac{\pi }{12})\right)+c$