Integrate the function: $x^{2} \log x$

(N/A) Let $I = \int x^{2} \log x \, dx$.
Using the integration by parts formula $\int u \cdot v \, dx = u \int v \, dx - \int \left( \frac{du}{dx} \int v \, dx \right) dx$,where we choose $u = \log x$ (first function) and $v = x^{2}$ (second function) based on the $LIATE$ rule:
$I = \log x \int x^{2} \, dx - \int \left( \frac{d}{dx}(\log x) \int x^{2} \, dx \right) dx$
$I = \log x \left( \frac{x^{3}}{3} \right) - \int \left( \frac{1}{x} \cdot \frac{x^{3}}{3} \right) dx$
$I = \frac{x^{3} \log x}{3} - \int \frac{x^{2}}{3} \, dx$
$I = \frac{x^{3} \log x}{3} - \frac{1}{3} \left( \frac{x^{3}}{3} \right) + C$
$I = \frac{x^{3} \log x}{3} - \frac{x^{3}}{9} + C$,where $C$ is an arbitrary constant.