Do we still add $C$ when integrating and the result is an arbitrary function?
When taking the antiderivative, anything that doesn't depend on the variable we're integrating over can be counted as a constant; for the space we're working with here, there's another variable that we can take into account when describing this "constant", so we can claim that any function in x is a valid potential solution. In this case, instead of calling it $f(x)$, I'd probably name it something like $C_y(x)$, so it's obvious where it came from: it's the constant of integration we got when integrating $u$ with respect to $y$.