It's my understanding that the definition of the indefinite integral is: ∫f(x)dx = F(x) + C, where d/dx [F(x) + C] = f(x) and C is an arbitrary constant And while dx has meaning apart from the indefinite integral sign the indefinite integral sign has no meaning apart from dx. Adding an indefinite integral sign to both sides of an equation would be similar to adding an open bracket to both sides of an equation; which of course is meaningless. So my question is when using the common method to solve separable diffeq why do we infact do this, e.g. dy/dx = xy dy/y = x dx ∫dy/y = ∫x dx ln [y] = x^2/2 + C y = exp[x^2/2 + C] My naive idea was that since all elementary antiderivatives can be written using other symbology besides the indefinite integral sign that adding an indefinite integral sign could be always be expressed as something else; I know this might seem silly but the method is nonsensical to begin with. However I think this is actually impossible to do because of very strange problems with the arbitrary constant.