Question

\[\int\limits_{\pi/6}^{\pi/4} cosec\ x\ dx\]

Answer 1

\[Let\ I = \int_\frac{\pi}{6}^\frac{\pi}{4} cosec x d x . Then, \]
\[I = \int_\frac{\pi}{6}^\frac{\pi}{4} cosec\ x \frac{cosec\ x - \cot x}{cosec x - \cot x} d x\]
\[ \Rightarrow I = \int_\frac{\pi}{6}^\frac{\pi}{4} \frac{{cosec}^2\ x - cosec\ x \cot x}{cosec\ x\ - \cot x} d x\]
\[ \Rightarrow I = \left[ \log \left( cosec\ x - \cot x \right) \right]_\frac{\pi}{6}^\frac{\pi}{4} \]
\[ \Rightarrow I = \log \left( \sqrt{2} - 1 \right) - \log\left( 2 - \sqrt{3} \right)\]