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I am trying to find a general solution in terms of $x$ and $f(x)$ of the integral

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

I tried partial integration, substitution and tried using the fact that $\frac{1}{f'(x)}$ is equal to $(f^{-1})'(y)$.

If the general integral does not exist, which criteria for $f$ do we have to add to make it solvable?

Edit: As Robert Israel pointed out, a "general solution" as described above does not exist. The question, for what $f$ one can formulate a closed-form solution, remains.

For instance, let us suppose that $f$ is a polynomial. Do we then get a closed-form solution?

Samuel
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    We know the antiderivative of any rational function, whether it's of the form $f(x)/f'(x)$ or not. – anon May 14 '20 at 00:42

1 Answers1

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There is certainly no simple expression for this in terms of $x$ and $f(x)$.

$$ \dfrac{d}{dx} F(x, f(x)) = F_1(x, f(x)) + f'(x) F_2(x, f(x))$$ where $F_1$ and $F_2$ are the partial derivatives of $F$. No way to get $f'$ in the denominator.

Of course, in particular cases you may be able to get closed-form expressions.

Robert Israel
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  • That is a nice explanation! Thank you! Even though it leads to the unsatisfactory result that I expected :/... do you know in what cases for $f$ I get a closed-form solution? – Samuel May 14 '20 at 00:31