Example 1. Find the integral $\int x e^{x^{2}} d x$.
Solution. Let $x^{2}=t$, then $2 x d x=d t, x d x=\frac{d t}{2}$. Substituting the obtained values into the integrand, we get

$$
\int x e^{x^{2}} d x=\int e^{x^{2}} x d x=\int e^{t} \frac{d t}{2}=\frac{1}{2} \int e^{t} d t=\frac{1}{2} e^{t}+C=\frac{1}{2} e^{x^{2}}+C
$$

This example can also be solved in another way (see § 4.2):

$$
\int x e^{x^{2}} d x=\int e^{x^{2}} x d x=\int e^{x^{2}} \frac{1}{2} d\left(x^{2}\right)=\frac{1}{2} \int e^{x^{2}} d\left(x^{2}\right)=\frac{1}{2} e^{x^{2}}+C
$$