Here is a little integral I made, but I think my solving steps could be flawed:
$$\int{f'(x)\cdot f(x)^{\left(\frac{f(x)}{\ln{f(x)}}\right)}dx}$$for $f(x)>1$.
My solution:
using $a^{\frac{1}{\ln(a)}}=e$,
$f(x)^{\left(\frac{f(x)}{\ln{f(x)}}\right)}=e^{f(x)}$, since $f(x)>1$ (This is where I think I'm wrong)
Therefore our integral becomes: $$\int{f'(x)\cdot e^{f(x)}dx}$$ $=e^{f(x)}+C$
Leaving us with a fairly nice answer. However, as mentioned before I feel as if there is something wrong with my first step. Im thinking that you aren't able to use that property with entire functions instead of just constants, right?
Is it actually correct? If not, why?
Another way I think one could see the same thing: Take $u=f(x)$: $$\int f’(x)\cdot f(x)^{\left(\frac{f(x)}{\ln f(x)}\right)}dx=\int u^{\left(\frac{u}{\ln u}\right) }du$$ Now use $z=\ln u\implies u=e^z$ and $du=e^z dz$. $$\int e^z\cdot\bigg(e^z\bigg)^{\left( \tfrac{e^z}{z}\right)}dz=\int e^{e^z} e^z dz=e^{e^z}+C$$$$=e^u+C=e^{f(x)}+C$$