Question

${\int }_0^{\pi /3}f\left(x\right)$ dx when f(x)= Minimum $\{tanx,cotx\}\forall x\in (0,\frac{\pi }{2})$

Answer 1

${\int }_0^{\Pi /3}f\left(x\right)dx$

$={\int }_0^{\Pi /3}Min\{tanx,cotx\}dx$

$={\int }_0^{\Pi /4}tanxdx+{\int }_{\Pi /4}^{\Pi /3}cotxdx$

$=Insec{\int }_0^{\Pi /4}+Insinx{\int }_{\Pi /4}^{\Pi /3}$

$=In\left(\sqrt{2}\right)-In\left(1\right)+In\left(\frac{\sqrt{3}}{2}\right)-In\left(\frac{1}{\sqrt{2}}\right)$

$=In\sqrt{2}-0+In\frac{\sqrt{3}}{2}+In\sqrt{2}$

$=2In\sqrt{2}+In\frac{\sqrt{3}}{2}=In{\left(\sqrt{2}\right)}^2+In\sqrt{3}-In2$

$=In2+In\sqrt{3}-In2$

$=In\sqrt{3}$