Q99 / 180 · COMEDK 2017 · Mathematics

Evaluate $\int \frac{1}{\sin x+\sqrt{3} \cos x} d x$.

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

Solution

We have, $\begin{aligned} \int \frac{1}{\sin x+\sqrt{3} \cos x} d x &=\frac{1}{2} \int \frac{d x}{\frac{1}{2} \sin x+\frac{\sqrt{3}}{2} \cos x} \\ &=\frac{1}{2} \int \frac{1}{\sin \left(x+\frac{\pi}{3}\right)} d x=\frac{1}{2} \int \operatorname{cosec}\left(x+\frac{\pi}{3}\right) d x \\ &=\frac{1}{2} \log \left|\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)\right|+C \end{aligned}$

Hence, the correct answer is option: 'C'.

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