Question

Evaluate : $\int\frac{1}{sin \,x +\sqrt{3} \,cos\, x} dx$

• A. $log \left|tan \left(\frac{x}{2}\right)\right| + C$
• B. $-\frac{1}{2}log \left|tan \left(\frac{x}{2}+\frac{\pi}{6}\right)\right| + C$
• C. $\frac{1}{2}log \left|tan \left(\frac{x}{2}+\frac{\pi}{6}\right)\right| + C$
• D. None of these

Answer 1

Answer: (C)

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

Answer 2

Let $I = \int\frac{1}{sin\, x +\sqrt{3} cos \,x} dx$ $= \frac{1}{2}\int\frac{dx}{\frac{1}{2}sin\, x + \frac{\sqrt{3}}{2} cos \,x}$ $\Rightarrow I= \frac{1}{2 }\int\frac{1}{sin \left(x+\frac{\pi}{3}\right)} dx = \frac{1}{2} \int cosec \left(x+\frac{\pi}{3}\right) dx$ $\Rightarrow I = \frac{1}{2}log \left|tan \left(\frac{x}{2}+\frac{\pi}{6}\right)\right| + C$