How should I integrate with logarithmic differential?

Question

I have a two variable function, f(x,y). I would like to integrate this function with respect to x in order for my result to be just as a function of y -- i.e $\int$ f(x,y) dx = f(y) . No problems here. However, I want to now transform this function into a log space, where essentially I would like to integrate over a logarithmic differential, i.e $\int$ f(x,y) dln(x) .

Is it a straightforward substitution, where I change dln(x) to du , and then every x in the function becomes e$^{u}$ ? I believe it is not this simple, however I'm a bit rusty on my integration techniques.

Answer 1

Try $$\int f(x,y) \text{d}(\ln(x))=\int f(x,y) \frac{\text{d}(\ln(x))}{\text{d}x} \text{d}x=\int f(x,y) \frac{\text{d}x}{x}.$$

Answer 2

The answer of Subrosar is valid. I would add that this is called the Stieltjes integral (Riemann-Stieltjes or Lebesgue-Stieltjes depending on the way you build the integral). As soon as $g' = \mathrm{d}g$ (the derivative is taken in the sense of distributions) is a measure, one can define $∫ f\,\mathrm{d}g$. Remark that here this is not the case (at least if you are in dimension $1$, so that you need more generally $∫ \frac{f(x)}{x}\,\mathrm{d}x$ to be well defined (so in particular, $f(0) = 0$. If $f(0)≠ 0$ but $f$ is $C^0$, then one can use distribution theory to get that $\mathrm{d}\ln(x)$ is the principal value of $1/x$ and so I would say that $$ \int_{\mathbb{R}} f(x,y)\,\mathrm{d}\ln(x) = \int \frac{f(x)-f(0)}{x}\,\mathrm{d} x. $$