calculus - Must a Function be Bounded for the Antiderivative to Exist over a Given Interval?

In my Calculus class, we were given the following definition of antiderivative:

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 of $f$ is denoted $\displaystyle \int f(x) dx$.

My question is, don't we have to say that $f$ should be bounded in the definition?

If not, then $f$ is not integrable by definition, so we can't say anything about $\displaystyle \int f(x) dx$, right?

I'm not sure if I'm thinking about this the right way.

Answer

The existence of an antiderivative and being integrable are distinct (although related) concepts.

Take

$$\begin{array}{rccc}f\colon&[0,1]&\longrightarrow&\mathbb R\\&&x\mapsto&\begin{cases}x^2\sin\left(\frac1{x^2}\right)&\text{ if }x>0\\0&\text{ if }x=0.\end{cases}\end{array}$$ Then $f$ is differentiable, but $f'$ is unbounded. But, in particular, $f$ is an antiderivative of $f'$.