Am I setting myself up for a fundamental misconception if I consider antidifferentiation denoted by $ \int $ sign as the operation that is the inverse of the differential of a function denoted by $ d $ and that also returns an arbitrary constant $ C $? To make things clear suppose I have a function $F$ where $\frac{d}{dx}F(x)=f(x)$ for all x in some interval. Then it follows $dF(x)=f(x)dx$ now if I have $\int (dF(x))=F(x)+C$ from my definition of antiderivative operation denoted by $\int$ then it also means $\int f(x)dx=F(x)+C$. What do you guys think?
Conventions for antiderivatives notation.
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integration
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0@Ilya, I thought the antiderivative was the inverse of the derivative, not the differential... I even posted a question precisely on this. Could you help please? http://math.stackexchange.com/questions/1035286/relationship-between-primitive-and-differential?noredirect=1#comment2112879_1035286 – 2014-11-24
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For a given (differentiable) function, there is another single function which is its derivative.
In the other direction we are not quite so lucky: if a function is integrable there is not a single function whose derivative is the original function, but a family of functions. Each member of the family corresponds to a particular choice of constant $C$.
So, when we say "the indefinite integral of f" and write "$\int f\ \ dx$" we are referring to the entire family at once. The notation "$F(x)+C$" is shorthand for $\{F(x)+C\mid C\in\mathbb{R}\}$.
If an initial condition is given, then it is possible to isolate exactly which member of the family you are looking for.