Answer
$$\ln 3.$$
Work Step by Step
We have
$$\int_{\pi/3}^{2\pi/3}\cot(\theta/2) d\theta=\int_{\pi/3}^{2\pi/3}\frac{\cos(\theta/2)}{\sin(\theta/2)} d\theta\\
=2\int_{\pi/3}^{2\pi/3}\frac{(\sin(\theta/2))'}{\sin(\theta/2)}=2\ln\sin(\theta/2)|_{\pi/3}^{2\pi/3}\\
=2(\ln\sin(\pi/3)-\ln \sin (\pi/6))=2\ln(\sqrt 3/2)-2\ln (1/2)=2\ln\sqrt 3=\ln 3.$$
