As I understand it, given some function $f$ integrable on $[a,b]$, we can define an indefinite integral of $f$ to be a function $F$ such that $$F(x)=\int_{n}^{x}f(t)dt,$$ where $n$ is some chosen fixed value on the $x$-axis such that $a \leq n \leq b$, and $x$ is a variable value varying between $n$ and $b$ inclusive.
We notice that aslong as $a \neq b$, there is an endless amount of different indefinite integral functions $F$ we can create for $f$, each one created by choosing a different value of $n$.
We call the set of all possible functions $F$ we can can create this way the indefinite integral of $f$, and denote it $$\int f(t)dt$$
Is my view of the indefinite integral correct? Am I on the right track? Or is there a fundamental misunderstanding somewhere in my explanation?
If this explanation is false, please explain why without assuming any knowledge of differential calculus; since the book I go through teaches integration before differentiation, I have only so far learnt about definite integrals.
Best Answer
That's close enough to right for your stage of learning. There are a few subtleties that can come later.
Doing integration first (which is where Archimedes started) you are setting yourself up for the fundamental theorem of calculus: when you learn about differentiation and then differentiate any one of those indefinite integrals you get the integrand.