Example 6. Calculate $\int_{1}^{e} \ln x d x$.
Solution. We apply the integration by parts formula (5.5).

Letting $u=\ln x, d v=d x$, we determine $d u=\frac{1}{x} d x, v=x$.

Therefore,

$$
\begin{gathered}
\int_{1}^{e} \ln x d x=\left.x \ln x\right|_{1} ^{e}-\int_{1}^{e} \frac{1}{x} x d x=\left.x \ln x\right|_{1} ^{e}-\int_{1}^{e} d x=\left.x \ln x\right|_{1} ^{e}-\left.x\right|_{1} ^{e}= \\
=e \ln e-1 \cdot \ln 1-(e-1)=e-0-e+1=1
\end{gathered}
$$