Answer:
$\int{ {x}^{2} \times \ln\left({{x}^{-1}}\right) } \mathrm{d} x$
$\int{ {x}^{2} \times \left( -\ln\left({x}\right) \right) } \mathrm{d} x$
$\int{ -{x}^{2} \times \ln\left({x}\right) } \mathrm{d} x$
$-\int{ {x}^{2} \times \ln\left({x}\right) } \mathrm{d} x$
$-\int{ \ln\left({x}\right) \times {x}^{2} } \mathrm{d} x$
$-\left( \ln\left({x}\right) \times \frac{ {x}^{3} }{ 3 }-\int{ \frac{ {x}^{3} }{ 3 } \times \frac{ 1 }{ x } } \mathrm{d} x \right)$
$-\left( \ln\left({x}\right) \times \frac{ {x}^{3} }{ 3 }-\int{ \frac{ {x}^{2} }{ 3 } } \mathrm{d} x \right)$
$-\left( \ln\left({x}\right) \times \frac{ {x}^{3} }{ 3 }-\frac{ 1 }{ 3 } \times \int{ {x}^{2} } \mathrm{d} x \right)$
$-\left( \ln\left({x}\right) \times \frac{ {x}^{3} }{ 3 }-\frac{ 1 }{ 3 } \times \frac{ {x}^{3} }{ 3 } \right)$
$-\frac{ \ln\left({x}\right) \times {x}^{3} }{ 3 }+\frac{ {x}^{3} }{ 9 }$
$\begin{array} { l }-\frac{ \ln\left({x}\right) \times {x}^{3} }{ 3 }+\frac{ {x}^{3} }{ 9 }+C,& C \in ℝ\end{array}$