My question deals with the definition of " indefinite integral" ( as conceptually distinguished from " primitive"). It may be the case that my question rests on some false assumption(s) ( for I'm new to the subject). In that case, I would be grateful these false assumptions to be pointed out.
I feel uncomfortable with the concept of "indefinite integral." I've once been answered on this site that "indefinite integral" means "more or less" the same thing as "primitive." But I'm not totally satisfied with this explanation, for (even though the 2 terms may denote the same set of objects) I cannot understand how the express the same concept.
Would it be possible to say that, for all functions $F$ and $f$ (defined on $I$): $$F=\int f(x)dx\iff F=\int_a^xf(t)dt\mbox{, for some $a\in I$}$$ in words, "$F$ is an indefinite integral of $f$ iff $F$ is an area function of $f$." Note: maybe a constant should be added to the area function?
Hence 2 questions:
Would the formula above suit as a definition of "indefinite integral?"
Or, in case it cannot count properly as a definition, is it at least a true statement? Does the "iff" really work in both senses, or only in one direction, or, in the worst case, is it false in both directions?
"Primitive" means anti-derivative, so this is a differential calculus term. "Integral" is an integral calculus term. The fact that the two coincide is called the Fundamental Theorem of Calculus.
Your attempted definition is almost a statement of the Fundamental Theorem so it won't work as a definition. Is it a true statement? It would be if you put some restrictions on $f$. Some functions are not integrate-able at all, so perhaps say that $f$ is piece-wise continuous on $I$.