$\newcommand{\d}{\mathrm{d}}$
Evaluate the integral using the indicated substituion. $$\int \cot x \csc^2x \,\d{x}, \qquad u= \cot x .$$
Differentiating both sides of $u$, then making the substitution: $$ \begin{align} u &= \phantom{-}\cot x, \\ \d u &= -\cot x\csc x \,\d{x}, \\ \d x &= -\frac{\d u}{u \csc x}. \end{align}$$ $$\int -\frac{u\csc^2 x \,\d{u}}{u\csc x} = \int -\csc x \,\d{u}. $$
Apparently, this was not an adequate approach, because $x$ is still part of the integrand. What should be done instead?