^ 3 _ 6 1 2x dx is equal to: - Vidyadip

MCQ

$\int\limits^\frac{\pi}{3}_\frac{\pi}{6}\frac{1}{\sin2\text{x}}\text{ dx}$ is equal to:

A
$\log_\text{e}{3}$
✓
$\log_\text{e}\sqrt{3}$
C
$\frac{1}{2}\log(-1)$
D
$\log(-1)$

✓

Answer

Correct option: B.

$\log_\text{e}\sqrt{3}$

$\int\limits^\frac{\pi}{3}_\frac{\pi}{6}\frac{1}{\sin2\text{x}}\text{ dx}$
$=\int\limits^\frac{\pi}{3}_\frac{\pi}{6}\operatorname{cosec}2\text{x}\text{ dx}$
$=\frac{1}{2}\int\limits^\frac{\pi}{3}_\frac{\pi}{6}2\operatorname{cosec}2\text{x}\text{ dx}$
$=\frac{-1}{2}\big[\log(\operatorname{cosec}2\text{x}+\cot2\text{x})\big]^\frac{\pi}{3}_\frac{\pi}{6}$
$=\frac{-1}{2}\big[-2\log\sqrt{3}\big]$
$=\log\sqrt{3}$

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