E x a m p l e 1. Find the indefinite integral $\int(2 x+3)^{2} d x$.
Solution. Based on the transformation of 3 differentials, we have

$$
d x=\frac{1}{2} d(2 x+3)
$$

Applying formula (9.5') for the case where $u=2 x+3, \alpha=2$, we find

$$
\int(2 x+3)^{2} d x=\int(2 x+3)^{2} \frac{1}{2} d(2 x+3)=\frac{1}{2} \int(2 x+3)^{2} d(2 x+3)=
$$

$$
=\frac{1}{2} \frac{(2 x+3)^{3}}{3}+C=\frac{1}{6}(2 x+3)^{3}+C
$$