Example 1.3. Compute $\int x e^{x} d x$.
Solution. Let $u=x, e^{x} d x=d v$. Then $d u=d x$, and $v=\int d v=\int e^{x} d x=e^{x}$ (it is not necessary to write the constant of integration here). By formula (1.10) we have

$$
\int x e^{x} d x=x e^{x}-\int e^{x} d x=x e^{x}-e^{x}+C=e^{x}(x-1)+C
$$