Example 2. Find the integral $\int x \sqrt{x-2} d x$.
Solution. To get rid of the root, let

$$
\sqrt{x-2}=t
$$

Squaring this equality, we find $x$ :

$$
x=t^{2}+2
$$

from which

$$
d x=2 t d t .
$$

Substituting the obtained equalities into the integrand, we find

$$
\begin{aligned}
\int x \sqrt{x-2} d x & =\int\left(t^{2}+2\right) t 2 t d t=\int\left(2 t^{4}+4 t^{2}\right) d t=2 \int t^{4} d t+4 \int t^{2} \cdot d t= \\
= & 2 \frac{t^{5}}{5}+4 \frac{t^{3}}{3}+C=\frac{2}{5}(x-2)^{\frac{5}{2}}+\frac{4}{3}(x-2)^{\frac{3}{2}}+C
\end{aligned}
$$