Example 9. Find $\int(2 x+3) \sqrt{x+2} d x$.
Solution. It is not obvious what to put under the differential sign, so let's make a substitution that allows us to get rid of the irrationality. Let $\sqrt{x+2}=t$, or $x+2=t^{2}$. From the last equality, we get $x=t^{2}-2, d x=2 t d t$, and $2 x+3=2\left(t^{2}-2\right)+3=$
$=2 t^{2}-1$. Substituting these equalities transforms the original integral into a new integral that can be reduced to tabular ones:

$$
\begin{aligned}
& \int(2 x+3) \sqrt{x+2} d x=\int\left(2 t^{2}-1\right) \cdot t \cdot 2 t d t= \\
& \quad=2 \int\left(2 t^{4}-t^{2}\right) d t=4 \frac{t^{5}}{5}-\frac{2}{3} t^{3}+C=\frac{4}{5} \sqrt{(x+2)^{5}}-\frac{2}{3} \sqrt{(x+2)^{3}}+C
\end{aligned}
$$

Thus, the substitution reduced to introducing the function $x=$ $=\varphi(t)=t^{2}-2$, and with respect to the variable $t$, the new integral became simpler.