1 +3 dx is equal to : - Vidyadip

MCQ

$\int\frac{1}{\cos\text{x}+\sqrt{3}\sin\text{x}}\text{ dx}$ is equal to :

A
$\log\tan\Big(\frac{\pi}{3}+\frac{\pi}{2}\Big)+\text{C}$
B
$\log\tan\Big(\frac{\pi}{2}-\frac{\pi}{3}\Big)+\text{C}$
✓
$\frac{1}{2}\log\tan\Big(\frac{\pi}{2}+\frac{\pi}{3}\Big)+\text{C}$
D
None of these.

✓

Answer

Correct option: C.

$\frac{1}{2}\log\tan\Big(\frac{\pi}{2}+\frac{\pi}{3}\Big)+\text{C}$

$\text{I}=\int\frac{1}{\cos\text{x}+\sqrt{3}\sin\text{x}}\text{ dx}$
$\text{I}=\frac{1}{2}\int\frac{2}{\frac{\cos\text{x}}{2}+\frac{\sqrt{3}}{2}\sin\text{x}}\text{ dx}$
$\text{I}=\frac{1}{2}\int\frac{1}{\cos\big(\text{x}-\frac{\pi}{6}\big)}\text{ dx}$
$\text{I}=\frac{1}{2}\int\sec\Big(\text{x}-\frac{\pi}{6}\Big)\text{dx}$
$\text{I}=\frac{1}{2}\ln\Big|\tan\Big(\frac{\text{x}}{2}+\frac{\pi}{3}\Big)\Big|+\text{C}$

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