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Question: Answered & Verified by Expert
$\int \frac{d x}{\cos x+\sqrt{3} \sin x}$ equals
MathematicsIndefinite IntegrationJEE MainJEE Main 2007
Options:
  • A
    $\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+c$
  • B
    $\frac{1}{2} \log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+c$
  • C
    $\log \tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)+\mathrm{c}$
  • D
    $\log \tan \left(\frac{\mathrm{x}}{2}-\frac{\pi}{12}\right)+\mathrm{c}$
Solution:
1002 Upvotes Verified Answer
The correct answer is:
$\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+c$
$\int \frac{d x}{\cos x+\sqrt{3} \sin x}$
$=\frac{1}{2} \int \sec \left(x-\frac{\pi}{3}\right) d x$
$=\frac{1}{2} \log \tan \left(\frac{x}{2}-\frac{\pi}{6}+\frac{\pi}{4}\right)+c$
$=\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+c$.

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