∫dxcos⁡x+3sin⁡x is equal to

# $\int \frac{\mathrm{dx}}{\mathrm{cos}\mathrm{x}+\sqrt{3}\mathrm{sin}\mathrm{x}}$ is equal to

1. A

$\frac{1}{2}\mathrm{log}\mathrm{tan}\left(\frac{\mathrm{x}}{2}+\frac{\mathrm{\pi }}{12}\right)+\mathrm{C}$

2. B

$\frac{1}{2}\mathrm{log}\mathrm{tan}\left(\frac{\mathrm{x}}{2}-\frac{\mathrm{\pi }}{12}\right)+\mathrm{C}$

3. C

$\mathrm{log}\mathrm{tan}\left(\frac{\mathrm{x}}{2}+\frac{\mathrm{\pi }}{12}\right)+\mathrm{C}$

4. D

$\mathrm{log}\mathrm{tan}\left(\frac{\mathrm{x}}{2}-\frac{\mathrm{\pi }}{12}\right)+\mathrm{C}$

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### Solution:

Now,

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

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