Question

Evaluate ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}}\csc xdx$

Answer 1

${\int }_{\left(\frac{\pi }{6}\right)}^{\left(\frac{\pi }{4}\right)}cosecxdx={[log|cosecx-cotx|]}_{\left(\frac{\pi }{6}\right)}^{\left(\frac{\pi }{4}\right)}=log\left|cosec\right(\frac{\pi }{4})-cot(\frac{\pi }{4}\left)\right|-log\left|cosec\right(\frac{\pi }{6})-cot(\frac{\pi }{6}\left)\right|=log|\sqrt{2}-1|-log|2-\sqrt{3}|=log\left(\right(\frac{\sqrt{2}-1}{2-\sqrt{3}}\left)\right)$