Question

Evaluate : $\int \frac{1}{sinx+\sqrt{3}cosx}dx$

• 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)

Let $I=\int \frac{1}{sinx+\sqrt{3}cosx}dx$

$=\frac{1}{2}\int \frac{dx}{\frac{1}{2}sinx+\frac{\sqrt{3}}{2}cosx}$

$\Rightarrow I=\frac{1}{2}\int \frac{1}{sin\left(x+\frac{\pi }{3}\right)}dx=\frac{1}{2}\int cosec\left(x+\frac{\pi }{3}\right)dx$

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