Integrate the function $x \sqrt{x+2}$.

Let $x+2=t$.
Then $x=t-2$ and $dx=dt$.
Substituting these into the integral:
$\int x \sqrt{x+2} \, dx = \int (t-2) \sqrt{t} \, dt$
$= \int (t \cdot t^{1/2} - 2t^{1/2}) \, dt$
$= \int (t^{3/2} - 2t^{1/2}) \, dt$
$= \int t^{3/2} \, dt - 2 \int t^{1/2} \, dt$
$= \frac{t^{5/2}}{5/2} - 2 \left( \frac{t^{3/2}}{3/2} \right) + C$
$= \frac{2}{5} t^{5/2} - \frac{4}{3} t^{3/2} + C$
Substituting $t = x+2$ back:
$= \frac{2}{5} (x+2)^{5/2} - \frac{4}{3} (x+2)^{3/2} + C$,where $C$ is the constant of integration.