I am trying to solve this integral
$$ \int \frac{f(x)}{g(x)}\frac{\mathrm dg}{\mathrm dx}\mathrm dx $$
where $g$ is an unknown function of $x$, and $f(x)$ is a known function that can be integrated or differentiated as necessary. Without $f$ the integral is just $\log(g(x))$, but I was wondering whether there is any opportunity for more progress.
I don't think integration by parts will work since in general $f(x)$ will not go to zero by repeated integration/differentiation.
Answer
As you might have noticed, we are integrating with respect to $g$. So you need to express $f(x)$ as a function of $g(x)$, i.e. $f(g(x)) = $ something to use regular integration methods. So because $f(x)$ is known, try expressing it as function of $g(x)$ and apply regular methods.
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