1. In the equation (*) both sides are infinitesimals.There is nothing wrong with this, but you must make sure that when one side of an equation that you write is infinitesimal so is the other.

2. An equation like , is a bit peculiar because, as we have already noted, A(x) is not uniquely determined by f and x; Thus the right hand side here is not a well defined function of x either.

3. Fortunately, we (next) deduce from the mean value theorem that all antiderivatives of the same function must differ from one another only by additives constants.Thus  is defined except for an additive constant. 

4. As a result of this, whenever you evaluate an antiderivative,you must add an undetermined constant to your answer. This reflects the fact that it is ungrammatical to give a well defined answer to a question that has no such answer. An antiderivative cannot be an ordinary function!