The Notes: `int_(pi/6)^(pi/2) cot^2(x) dx` Evaluate the integral

$\displaystyle{\int_{{\frac{\pi}{{6}}}}^{{\frac{\pi}{{2}}}}}{{\cot}}^{{2}}{\left({x}\right)}{\left.{d}{x}\right.}$

Let's first evaluate the indefinite integral,

Use the identity: $\displaystyle{{\cot}}^{{2}}{\left({x}\right)}={{\csc}}^{{2}}{\left({x}\right)}-{1}$

$\displaystyle\int{{\cot}}^{{2}}{\left({x}\right)}{\left.{d}{x}\right.}=\int{\left({{\csc}}^{{2}}{\left({x}\right)}-{1}\right)}{\left.{d}{x}\right.}$

$\displaystyle=\int{{\csc}}^{{2}}{\left({x}\right)}{\left.{d}{x}\right.}-\int{1}{\left.{d}{x}\right.}$

use the common integral: $\displaystyle\int{{\csc}}^{{2}}{\left({x}\right)}{\left.{d}{x}\right.}=-{\cot{{\left({x}\right)}}}$

$\displaystyle=-{\cot{{\left({x}\right)}}}-{x}$

$\displaystyle\therefore{\int_{{\frac{\pi}{{6}}}}^{{\frac{\pi}{{2}}}}}{{\cot}}^{{2}}{\left({x}\right)}{\left.{d}{x}\right.}={{\left[-{\cot{{\left({x}\right)}}}-{x}\right]}_{{\frac{\pi}{{6}}}}^{{\frac{\pi}{{2}}}}}$

$\displaystyle={\left[-{\cot{{\left(\frac{\pi}{{2}}\right)}}}-\frac{\pi}{{2}}\right]}-{\left[-{\cot{{\left(\frac{\pi}{{6}}\right)}}}-\frac{\pi}{{6}}\right]}$