Question

$\underset{\frac{\pi }{4}}{\overset{{}^{\frac{\pi }{3}}}{\int }}\cot xdx$

Answer 1

$I={\int }_{\pi /4}^{\pi /3}\cot xdx$
$={\int }_{\pi /4}^{\pi /3}\frac{\cos x}{\sin x}dx$
$={\left[\ln \left|\sin x\right|\right]}_{\pi /4}^{\pi /3}$
$=\ln \left(\sin \pi /3\right)-\ln \left(\sin \pi /4\right)$
$=\ln \left(\frac{\sqrt{3}}{2}\right)-\ln \left(\frac{1}{\sqrt{2}}\right)$
$=\ln \left(\frac{\sqrt{3}}{\sqrt{2}}\right)$
$\therefore$Answer: $\ln \left(\frac{\sqrt{3}}{\sqrt{2}}\right)$