Question

Evaluate the definite integral ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}}co\sec xdx$

Answer 1

Let $I={\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}}co\sec xdx$
$\Rightarrow \int co\sec xdx=\log |co\sec x-\cot x|=F\left(x\right)$
By second fundamental theorem of calculus, we obtain
$I=F\left(\frac{\pi }{4}\right)-F\left(\frac{\pi }{6}\right)$
$=\log |co\sec \frac{\pi }{4}-\cot \frac{\pi }{4}|-\log |co\sec \frac{\pi }{6}-\cot \frac{\pi }{6}|$
$=\log |\sqrt{2}-1|-\log |2-\sqrt{3}|$
$=\log \left(\frac{\sqrt{2}-1}{2-\sqrt{3}}\right)$