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For my current research in economics: my result depends crucially on but one integral: $$\int\ln(f(x))\,\mathrm{d}x$$ But I know very little (practically nothing) about the shape of function $f$. How can I integrate this? Is there a general form for the solution? I can provide more background if needed.

frencho
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    There is no general form, more information on $f$ is needed. – Delta-u Apr 16 '18 at 16:09
  • Thanks. I found elsewhere that I could integrate by parts, and that I would get $\int\ln(f(x))dx=x\ln(f(x))-\int\frac{xf'(x)}{f(x)}dx$ Anyone confirms? – frencho Apr 16 '18 at 16:15
  • Yes that is correct – The Integrator Apr 16 '18 at 16:16
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    @frencho This isn't really a reasonable question. Your question only makes sense for nonnegative $f$, so we can write $f=e^{g}$. Then your question reduces to how to integrate $$\int g(x)~\mathrm d x$$ for some real function $g$. This is too general for a question here. – user159517 Apr 16 '18 at 16:18
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    Great. And the last integral in this result cannot be computed unless I know what $f(x)$ is, correct? – frencho Apr 16 '18 at 16:18
  • @frencho I don't see how you could expect to calculate any integral involving $f$ without some assumptions on $f$ (e.g., like $f$ being polynomial) – user159517 Apr 16 '18 at 16:20
  • @user159517. Point taken. Indeed from the very few things I know about my function $f$ is that it cannot be negative-valued. And yes if it helps $f$ is some polynomial in $x$, but I don’t know which polynomial it is. – frencho Apr 16 '18 at 16:20
  • @frencho well, that's an assumption you need for the integral even to be defined. Maybe you could add some background to this question and explain the context. – user159517 Apr 16 '18 at 16:23
  • Many thanks to all of you good folks. Very rusty with my math, so my apologies. Not to sidestep some questions, but I just dont know the answers. But let's go back to $$\int\ln(f(x))dx=x\ln f(x)-\int\frac{xf'(x)}{f(x)}dx$$ for a moment because I think this is very useful for my purposes. next question: Consider the last integral $$\int\frac{xf'(x)}{f(x)}dx$$. I'm not sure what this amounts to. But I think that for my purposes I can make some progress by making a variable change so that I would integrate with respect to time $t$ – frencho Apr 16 '18 at 16:36
  • (I suspect something is going on with rates of change over time). – frencho Apr 16 '18 at 16:36
  • in other words, with a proper variable change, isn't $f'(x)/f(x)$ some kind of rate of growth? – frencho Apr 16 '18 at 16:38

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You could try integrating by parts:

$$\int ln(f(x))dx=\int 1\cdot ln(f(x))dx= xln(f(x))-\int x\frac{f'(x)}{f(x)}dx$$

You can then continue depending on what you know about the function f(x).

Anastassis Kapetanakis
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  • Awesome. Confirmed twice! But I know next to nothing regarding $f$, however whatever math I have left in me senses that something is going on with the last integral, maybe some kind of rate of change over time if we do a variable change from integrating wrt $x$ to integrating wrt $t$ (time). Is there a hint of truth in this intuition? – frencho Apr 16 '18 at 16:43