Integrate the function: $x \log(2x)$

Let $I = \int x \log(2x) \, dx$.
Using the integration by parts formula $\int u v \, dx = u \int v \, dx - \int (u' \int v \, dx) \, dx$,where $u = \log(2x)$ and $v = x$.
Then $u' = \frac{1}{2x} \cdot 2 = \frac{1}{x}$ and $\int v \, dx = \frac{x^2}{2}$.
Substituting these into the formula:
$I = \log(2x) \cdot \frac{x^2}{2} - \int \left( \frac{1}{x} \cdot \frac{x^2}{2} \right) \, dx$
$I = \frac{x^2 \log(2x)}{2} - \int \frac{x}{2} \, dx$
$I = \frac{x^2 \log(2x)}{2} - \frac{1}{2} \cdot \frac{x^2}{2} + C$
$I = \frac{x^2 \log(2x)}{2} - \frac{x^2}{4} + C$,where $C$ is an arbitrary constant.