Q84 / 150 · MHT CET 2021 (22 Sep Shift 2) · Mathematics

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

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

Solution

Dividing numerator and denominator by 2 , we get $$ =\frac{1}{2} \int \frac{\mathrm{dx}}{\left(\frac{1}{2} \cos \mathrm{x}+\frac{\sqrt{3}}{2} \sin \mathrm{x}\right)}=\frac{1}{2} \int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}+\frac{\pi}{6}\right)}=\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)\right|+\mathrm{C} $$

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

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