I'm trying to figure out if the following definition of antiderivative is correct:
Let $f$ be a function defined on an interval. An antiderivative of $f$ is any function $F$ such that $F' = f$. The collection of all antiderivatives is denoted $\displaystyle \int f(x) dx$.
Is it okay to have no conditions on $f$ that ensure it's integrable?
Let $$F(x)=\begin{cases} x^2\sin (1/x^2) &\text{ if } 0<x\leq 1\\ 0 & \text{ if } x=0\end{cases}$$ You can check that $F$ is differentiable on $[0,1]$. Set $f:=F'.$ Note that $F'$ is not bounded on $[0,1]$, so it is not Riemann integrable on $[0,1]$. Thus, $f$ has an anti-derivative, namely $F$, but it is not Riemann integrable.