Example 2. Find $\int x \ln x d x$.
Solution. Let

$$
\ln x=u, x d x=d v
$$

we get

$$
\frac{1}{x} d x=d u, \frac{x^{2}}{2}=v
$$

By formula (4.18) we find

$$
\int x \ln x d x=\frac{x^{2}}{2} \ln x-\int \frac{x^{2}}{2} \cdot \frac{1}{x} d x=\frac{x^{2}}{2} \ln x-\frac{1}{4} x^{2}+C
$$