Q54 / 120 · VITEEE 2015 · Mathematics

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

  • Correct answer
    A
    $\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+C$
  • B
    $\frac{1}{3} \log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+C$
  • C
    $\log \tan \left(\frac{x}{2}+\frac{\pi}{6}\right)+C$
  • D
    $\frac{1}{2} \log \tan \left(\frac{x}{2}-\frac{\pi}{6}\right)+C$

Solution

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

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

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