What’s the formal definition of an indefinite integral

calculusdefinitionindefinite-integralsintegrationreal-analysis

I understand that definite integrals can be defined as the limit of Riemann sums but I have yet to see a similar definition for the indefinite integral. I suppose
$$
\int f(x) \, dx = \{F(x):F'=f \}
$$
might do the trick but I'm unsure if this really a satisfactory definition. Alternatively, with the knowledge of the fundamental theorem of calculus, we might define
$$
\int f(x) \, dx = \int_{a}^{x} f(t) \, dt + C
$$
but this definition seems odd in that uses quite a deep result to define something which is usually introduced very early on in calculus classes. So is there a formal definition of an indefinite integral, and if so, what is it?

Best Answer

You should stick to the definition of an indefinite integral.

Given a function $f:I\to\mathbb{R}$ defined on an open interval $I$, if $F:I\to\mathbb{R}$ is a function such that $F'(x)=f(x)$ for every $x\in I$, then we call $F$ an antiderivative of the function $f$. It is easy to check the two following two facts:

  • If $F$ is an antiderivative $f$ on the interval $I$, then so is $F+C$ for any constant $C$;

  • If $F$ and $G$ are two antiderivatives of $f$, i.e, $F'(x)=G'(x)=f(x)$ for all $x\in I$, then there exists (exercise!) a constant $C$ such that $F=G+C$.

We define the notion of the "indefinite integral" of the function $f$ (on the interval $I$) as a family of functions:

$$ \textrm{indefinite integral of } f=\{F:I\to\mathbb{R}\mid F'(x)=f(x)\ \textrm{for all } x\in I\} $$

Because of these two facts above, we write $$ \int f(x)\,dx=F(x)+C $$ for any antiderivative $F$ of $f$ (on the interval $I$).

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