Answer
$$0$$
Work Step by Step
$$\eqalign{
& \int_\pi ^{2\pi } {\cot \frac{x}{3}} dx \cr
& = \int_\pi ^{2\pi } {\left( 3 \right)\cot \frac{x}{3}\left( {\frac{1}{3}} \right)} dx \cr
& {\text{find the antiderivative}} \cr
& = 3\left( {\ln \left| {\sin \frac{x}{3}} \right|} \right)_\pi ^{2\pi } \cr
& {\text{evaluate limits}} \cr
& = 3\left( {\ln \left| {\sin \frac{{2\pi }}{3}} \right| - \ln \left| {\sin \frac{\pi }{3}} \right|} \right) \cr
& {\text{simplify}} \cr
& = 3\left( {\ln \left| {\frac{{\sqrt 3 }}{2}} \right| - \ln \left| {\frac{{\sqrt 3 }}{2}} \right|} \right) \cr
& = 0 \cr} $$