Can the expression be represented as a total differential?

I should determine whether the following expression represents a total differential:


$$\mathrm{d}f(x,y) = (\ln y) \,\mathrm{d}x + \frac{x}{y}\,\mathrm{d}y$$


But with the following method (physical chemistry):



> 
> Prove your statements assuming the existence of a "state function" $f$ and integration of the total differential. If necessary, give the function $f(x,y)$.
> 
> 
> Note: Start with the definition of a total differential and **do not** use Schwarz's theorem.
> 
> 
> 


So first I looked at the definition of a total differential.



> 
> In general, if the integral over an infinitesimal quantity is
> independent of the path between the integration limits, this quantity
> is called **total differential**. ... In general, if the integral over an
> infinitesimal quantity depends on the path between the integration
> limits, this quantity is **not called a total differential**. - Atkins,
> Physical Chemistry, 6th edition
> 
> 
> 


So if I now minintegrate $\mathrm{d}f(x,y)$ first as suggested, I arrive at:


$$f(x, y) = 2x\ln y$$


$f(x,y)$ could now be, for example, my internal energy $U$, $x$ my temperature and $y$ my volume $V$. Then the function would give me the change in $U$, since the following applies:


$$\Delta U = \int \mathrm{d}U$$


I have several more tasks of this kind here, could I not then arbitrarily say that is $V$, that is $T$, that is $U$ and therefore $U$ is always independent of the way, so all functions are total differentials? I somehow do not know what to do.


