# How to solve $\int \frac{f(x)}{f^{\prime}(x)} dx$

Let $$f:\mathbb{R}\to\mathbb{R}$$ be of class $$C^{\infty}$$ with $$f^{\prime}\not\equiv0$$.

There exists a formula to solve the integral $$\int \frac{f(x)}{f^{\prime}(x)} dx?$$

Since I know that is $$\int \frac{f^{\prime}(x)}{f(x)} dx=\log(|f(x)|) +c,$$

I was wondering how to act when in the first case.

I hope someone could help. Thank you in advance.

• For an arbitrary function $h$ and $f = e^h$ that would be $\int \frac{dx}{h'(x)}$, therefore I doubt that there is a general formula. Feb 2 at 13:17
• @MartinR You observation reduces to $[(\log f)']^{-1}={f\over f'}.$ Feb 2 at 13:29
• There is no known formula for this. Feb 2 at 13:51

$$\int \frac{f(x)}{f'(x)} dx$$ can not be expressed in terms of $$f$$ and elementary functions of $$f$$ for arbitrary functions $$f$$, as this example demonstrates:
Consider $$f(x) = e^{x \log x - x}$$ for $$x > 1$$. Then $$\frac{f(x)}{f'(x)} = \frac{1}{\log(x)}\, ,$$ but that has no antiderivative in terms of elementary function, see for example Integral of $\int\frac{1}{\log x}dx$ (and the references therein).
• then for which kind of nontrivial (I mean e.g. $f(x)=x$) functions it is possible to evaluate that integral? Feb 2 at 16:30
• If $f$ is rational then $f/f'$ is also rational. Similar for rational functions of $\sin(x)$ and $\cos(x)$. I am not aware of a general characterisations of functions $f$ for which $f/f'$ is elementary integrable. Feb 2 at 19:59