\[
\int\ln x dx = \int 1\cdot\ln x dx\ \ \ \ \ \ \ \ \
\begin{align}
u(x)&=\ln x\ &\ \dot{u}(x)=x^{-1}\\
\dot{v}(x)&=1\ &\ v(x)=x\\
\end{align}
\]
\[
\int\ln x dx=\ln x \cdot x-\int x^{-1}\cdot x dx = \ln x \cdot x - x + c
\]