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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.

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    $\begingroup$ 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. $\endgroup$
    – Martin R
    Feb 2 at 13:17
  • $\begingroup$ @MartinR You observation reduces to $[(\log f)']^{-1}={f\over f'}.$ $\endgroup$
    – Ryszard Szwarc
    Feb 2 at 13:29
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    $\begingroup$ There is no known formula for this. $\endgroup$
    – GEdgar
    Feb 2 at 13:51

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$\int \frac{f(x)}{f'(x)} dx$ 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).

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  • $\begingroup$ then for which kind of nontrivial (I mean e.g. $f(x)=x$) functions it is possible to evaluate that integral? $\endgroup$
    – C. Bishop
    Feb 2 at 16:30
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    $\begingroup$ 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. $\endgroup$
    – Martin R
    Feb 2 at 19:59

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