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

$\int \frac{1}{\cos x+\sqrt{3} \sin x} d x=$
  1. $2 \log \left[\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)\right]+\mathrm{c}$
  2. $\frac{1}{2} \log \left[\tan \left(\frac{x}{2}-\frac{\pi}{12}\right)\right]+c$
  3. $\frac{1}{2} \log \left[\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)\right]+\mathrm{c}$
  4. $2 \log \left[\tan \left(\frac{x}{2}-\frac{\pi}{12}\right)\right]+c$

Solution

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

Asked in: MHT CET 2021 (22 Sep Shift 2)

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