Example 1. Find $\int e^{x^{2}} \cdot x d x$.
Solution. Since $\int e^{u} d u=e^{u}+C$, in this integral, it is necessary to make $x^{2}$ the variable of integration, i.e., let $x^{2}=u$. In this case, $d u=d x^{2}=2 x d x$. Therefore, we bring the factor $x$ under the differential sign and obtain the differential of $x^{2}$: $x d x=\frac{1}{2} d\left(x^{2}\right)$. Then

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