Question

\[\int\limits_{\pi/6}^{\pi/3} \frac{1}{\sin 2x} dx\]  is equal to

Options

•  log_e 3

• \[\log_e \sqrt{3}\]
• \[\frac{1}{2}\log\left( - 1 \right)\]

•  log (−1)

   

Answer 1

\[\log_e \sqrt{3}\]

\[\int_\frac{\pi}{6}^\frac{\pi}{3} \frac{1}{\sin2x} d x\]
\[ = \int_\frac{\pi}{6}^\frac{\pi}{3} \ cosec2x\ dx\]
\[ = \frac{1}{2} \int_\frac{\pi}{6}^\frac{\pi}{3} 2\ cosec2x\ dx\]
\[ = \frac{- 1}{2} \left[ \log\left( \ cosec\ 2x\ + \cot2x \right) \right]_\frac{\pi}{6}^\frac{\pi}{3} \]
\[ = \frac{- 1}{2}\left[ - 2\log\sqrt{3} \right]\]
\[ = \log\sqrt{3}\]