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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.

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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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  • $\begingroup$ This is fine if $g$ is one to one, but wouldn't you run into problems if it wasn't? Also, I don't know what $g$ is, if possible I'd like to leave it general. $\endgroup$ – David May 27 '15 at 3:02
  • $\begingroup$ @David not in this case. We are not forcing another function on $x$, we are just rewriting in the form $\int \frac{f(g)}{g} dg$. We would need a one-to-one function if we had limits of integration or if we had to put the result back in terms of x, because we would need to guarantee that the limits were fine and we could transfer back to the x. Ask me again if you don't understand what I just said above. $\endgroup$ – SalmonKiller May 27 '15 at 3:05
  • $\begingroup$ OK I see. It's not as good as I'd hoped it might be, since I still can't do the resulting integral, but it is a start, and realistically I can't see it working out much better. If I don't get a better answer I'll accept yours in due time. $\endgroup$ – David May 27 '15 at 3:23

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