A version of Liouville's criterion states that: Choose functions $~f~$ and $~g~$ such that $~f≠0~$ and $~g~$ is non-constant. Then, the function $~f(x)e^{g(x)}~$ can be integrated in elementary terms iff there exits a rational function $~R(x)~$ such that $$~R'(x)+g'(x)R(x)=f(x)~.$$ I think I may have used the criteria wrongly, since it gives contradictory results?
I considered the function $~e^x~$, which obviously has an elementary anti-derivative. Then, I let $~f=1~$ and $~g=x~$, so the $~1^{st}~$ order differential equation I had to solve was $~R'(x)+R(x)=1~$.
The general solution to this D.E. is $~R(x)=Ce^{-x}+1~$, where $~c~$ is an arbitrary constant.
However, $~R(x)~$ is clearly not a rational function, so I don't know where my misconception lies.
Is it because Louville's criteria considers complex-valued functions of the real variable $~x~$?
