What is the difference between an indefinite integral and an antiderivative?

I thought these were different words for the same thing, but it seems I am wrong. Help.

Solution 1:

An anti-derivative of a function $f$ is a function $F$ such that $F'=f$.

The indefinte integral $\int f(x)\,\mathrm dx$ of $f$ (that is, a function $F$ such that $\int_a^bf(x)\,\mathrm dx=F(b)-F(a)$ for all $a<b$) is an antiderivative if $f$ is continuous, but need not be an antiderivative in the general case.

Solution 2:

"Indefinite integral" and "anti-derivative(s)" are the same thing, and are the same as "primitive(s)".

(Integrals with one or more limits "infinity" are "improper".)

Added: and, of course, usage varies. That is, it is possible to find examples of incompatible uses. And, quite seriously, $F(b)=\int_a^b f(t)\,dt$ is different from $F(x)=\int_a^x f(t)\,dt$ in what fundamental way? And from $\int_0^x f(t)\,dt$? And from the same expression when $f$ may not be as nice as we'd want?

I have no objection if people want to name these things differently, and/or insist that they are somewhat different, but I do not see them as fundamentally different.

So, the real point is just to be aware of the usage in whatever source...

(No, I'd not like to be in a classroom situation where grades hinged delicately on such supposed distinctions.)