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

`int_(pi/6)^(pi/2)cot^2(x)dx`

Let's first evaluate the indefinite integral,

Use the identity:`cot^2(x)=csc^2(x)-1`

`intcot^2(x)dx=int(csc^2(x)-1)dx`

`=intcsc^2(x)dx-int1dx`

use the common integral: `intcsc^2(x)dx=-cot(x)`

`=-cot(x)-x`

`:.int_(pi/6)^(pi/2)cot^2(x)dx=[-cot(x)-x]_(pi/6)^(pi/2)`

`=[-cot(pi/2)-pi/2]-[-cot(pi/6)-pi/6]`

`=[-pi/2]-[-sqrt(3)-pi/6]`

`=-pi/2+sqrt(3)+pi/6`

`=sqrt(3)-pi/3`