Integration of $\frac{f'(x)}{f(x)}$? $\frac{f'(x)}{f(x)}$ integrated must be
$$\int\frac{f'(x)}{f(x)}dx=\ln\rvert f(x)\rvert+c.$$
But when trying to show this with partial integration I get another result:
$$
\begin{align*}
&\int\frac{\frac{d}{dx}f(x)}{f(x)}dx \\
=\quad&\int\frac{d}{dx}(f(x))\cdot (f(x))^{-1}dx \\
=\quad&f(x)\cdot (f(x))^{-1}-\int -f(x)\cdot (f(x))^{-2}dx \\
=\quad&1+\int (f(x))^{-1}dx \\
=\quad&1+\ln\rvert f(x)\rvert+c
\end{align*}
$$
Is there a mistake or can I summarize $1+c$, because $1$ is a constant?

Any function of arbitrary constants is also another arbitrary constant.
