# How to solve/rewrite $\int\ln(f(x))dx$ when $x$ is a function of time $t$

OK, I've made some major progress on an important paper of mine and it all boils down to solving/rewriting the integral $\int\ln(f(x))dx$ but the twist is that $x$ is some unknown function of $t$, i.e. $$\int\ln(f(x(t)))dx(t)$$ and I'm looking for some sort of general solution, because I do not have an expression for $f$ or $x(t)$. I just want to rewrite this thing as a function of $x(t)$ or $t$. What would you do? Smells like variable change to me, but my poor brain is melting. I do know, however, that $$\int\ln(f(x))dx=x\ln f(x)-\int\frac{xf'(x)}{f(x)}dx$$ but I'm unsure what to do with this once I allow for $x$ to be $x(t)$. Any ideas? I can provide more info.

• How exactly do you want to write the integral? Are there special needs for what you are trying to show?
– ℋolo
Apr 16 '18 at 18:54
• Please give us any information you've got for the function f(x)! Apr 16 '18 at 18:58
• Much thanks, and sorry to be not clear enough, so let me try again. I need to rewrite $$\int\ln(f(x(t)))dx(t)$$ as function of $x(t)$, or its absolute change $\Delta x(t)$, or possibly its relative change $\Delta x(t)/x(t)$. Again I don't know anything about $f$, it's not a numerical derivation I'm looking for. The only other information I have is that $x(t)\in[0,1]$, as well as the decomposition which I repeat: $$\int\ln(f(x))dx=x\ln f(x)-\int\frac{xf'(x)}{f(x)}dx$$. Again I'm looking to rewrite this using $x(t)$, $\Delta x(t)$, or $\Delta x(t)/x(t)$. Apr 16 '18 at 20:44
• Is this feasible, or impossible without knowing $f$. Might be a dumb question, but not to me. Apr 16 '18 at 20:45

Does it matter? You are still integrating the function $x(t)$ with respect to itself.
It's a bit like integrating $y^2$ with respect to $y$ but we have that $y=x^3$. Either way you would get $y^3/3=x^9/3$.