Mathematics | VITEEE 2015 | Discussion Page

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

$\frac{1}{2}\log \tan (\frac{x}{2}+\frac{\pi}{12})+C$

$\frac{1}{3}\log \tan (\frac{x}{2}-\frac{\pi}{12})+C$

$\log \tan (\frac{x}{2}+\frac{\pi}{6})+C$

$\frac{1}{3}\log \tan (\frac{x}{2}-\frac{\pi}{6})+C$

Option A

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

= $\frac{1}{2}\int \frac{dx}{\frac{1}{2}\cos x+\frac{\sqrt{3}}{2}sin x}$

= $\frac{1}{2}\int \frac{dx}{\cos\frac{\pi}{3}\cos x+\sin\frac{\pi}{3}sin x}$

= $\frac{1}{2}\int \frac{dx}{\cos(x-\frac{\pi}{3})}$

= $\frac{1}{2}\int \sec (x-\frac{\pi}{3}) dx$

= $\frac{1}{2}\log \tan (\frac{x}{2}-\frac{\pi}{6}+\frac{\pi}{4})+C$

= $\frac{1}{2}\log \tan (\frac{x}{2}+\frac{\pi}{12})+C$

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