Here, the "constant" of integration is no longer a constant, but instead a function of all the variables of the original function except "x". The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve formula_2 will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative. The most general way to represent this is to have the "constant" represent an unknown function of all the other variables.