Antiderivative of the natural logarithm
We wish to determine the antiderivative $\int \ln(x)\, dx$. We can solve this with integration by parts by rewriting the integrand as $1\cdot \ln(x)$, and using the facts that $\int 1 \,dx = x$ and $\left(\ln(x)\right)'=\frac{1}{x}$: \begin{align} \int \ln(x)\, dx &= \int 1\cdot \ln(x)\, dx\\ &= \int 1\, dx\cdot \ln(x) - \int \left(\int 1\, dx\right) \left(\ln(x)\right)'\, dx \\ &= x \ln(x) - \int x \cdot \frac{1}{x}\,dx \\ &= x \ln(x) - \int 1 \,dx \\ &= x \ln(x) - x\,, \end{align} and we've recovered the familar expression.
