Answer
$$\frac{4}{3}-\ln \sqrt{3}$$
Work Step by Step
We integrate as follows:
\begin{align*}
\int_{\pi / 6}^{\pi / 3} \cot ^{3} x d x&=\int_{\pi / 6}^{\pi / 3}\left(\csc ^{2} x-1\right) \cot x d x\\
&=\int_{\pi / 6}^{\pi / 3} \csc ^{2} x \cot x d x-\int_{\pi / 6}^{\pi / 3} \cot x d x\\
&=\left(-\frac{\cot ^{2} x}{2}+\ln |\csc x|\right)\bigg|_{\pi / 6}^{\pi / 3}\\
&=-\frac{1}{2}\left(\frac{1}{3}-3\right)+\left(\ln \frac{2}{\sqrt{3}}-\ln 2\right)\\
&=\frac{4}{3}-\ln \sqrt{3}
\end{align*}
