Integrate the function: $x \log x$

Let $I = \int x \log x \, dx$.
Using the integration by parts formula: $\int u \cdot v \, dx = u \int v \, dx - \int \left( \frac{du}{dx} \int v \, dx \right) dx$.
Taking $u = \log x$ (first function) and $v = x$ (second function) based on the $LIATE$ rule:
$I = \log x \int x \, dx - \int \left( \frac{d}{dx} \log x \cdot \int x \, dx \right) dx$
$I = \log x \cdot \frac{x^2}{2} - \int \left( \frac{1}{x} \cdot \frac{x^2}{2} \right) dx$
$I = \frac{x^2 \log x}{2} - \int \frac{x}{2} \, dx$
$I = \frac{x^2 \log x}{2} - \frac{1}{2} \cdot \frac{x^2}{2} + C$
$I = \frac{x^2 \log x}{2} - \frac{x^2}{4} + C$,where $C$ is the constant of integration.