Answer
$$6\sqrt 3 - 2\pi $$
Work Step by Step
$$\eqalign{
& \int_\pi ^{3\pi } {{{\cot }^2}\frac{x}{6}} dx \cr
& {\text{Use the trigonometric identity}}\cr
& \cot^2 \theta + 1 = {\csc ^2}\theta \cr
& = \int_\pi ^{3\pi } {\left( {{{\csc }^2}\frac{x}{6} - 1} \right)} dx \cr
& {\text{We know that }}\frac{d}{{dx}}\left[ {\cot x} \right] = - {\csc ^2}x.{\text{ Then}}{\text{,}} \cr
& = \left( { - 6\cot \left( {\frac{x}{6}} \right) - x} \right)_\pi ^{3\pi } \cr
& {\text{Evaluate}} \cr
& = \left( { - 6\cot \left( {\frac{{3\pi }}{6}} \right) - 3\pi } \right) - \left( { - 6\cot \left( {\frac{\pi }{6}} \right) - \pi } \right) \cr
& {\text{Simplifying}} \cr
& = - 6\left( 0 \right) - 3\pi + 6\sqrt 3 + \pi \cr
& = 6\sqrt 3 - 2\pi \cr} $$
