Undergrad Ambiguity of the term "indefinite integral"

[u0x2a]

TL;DR

The sense in which the term is used is often unclear to me from the context

I've encountered a few different definitions of "indefinite integral," denoted $\int f(x) \, dx$.

1. any particular antiderivative $F:\mathbb{R} \to \mathbb{R}, F'(x) = f(x)$
2. the set of all antiderivatives $\{F:\mathbb{R} \to \mathbb{R}, F'(x) = f(x)\}$
3. a "canonical" antiderivative
4. any expression of the form $\int_a^x f(x) \, dx$, where $a$ is in the domain of $f$ and $f$ is continuous

Sometimes, it becomes a little unclear which definition an author really has in mind, though, when they start proving properties of "the indefinite integral." For example, authors often say "the indefinite integral is linear", ie:
$$\int \left( af + bg \right) \, dx = a \int f \, dx + b \int g \, dx$$

If I interpret this statement in terms of particular antiderivatives, then it's true for all antiderivatives $\int \left( af + bg \right) \, dx$, $\int f \, dx$, $\int g \, dx$, but only up to a constant, which isn't mentioned.

If I interpret this statement in terms of sets of antiderivatives, then it's also true but it uses addition and multiplication in ways that were never defined, involving both scalars and equivalence classes of functions that have the same derivative.

If I interpret this statement in terms of "canonical antiderivatives", then I'm not sure it's "even wrong" or "even right" because "canonical antiderivative" isn't fully defined or even shown to exist. I can imagine defining a canonical antiderivative by defining a partial inverse of the differentiation operator $\frac{d}{dx}$ in the same way we define a partial inverse of the square function. We can make $\frac{d}{dx}$ surjective by restricting the codomain to functions that have antiderivatives. But then to make $\frac{d}{dx}$ injective, we have to restrict the domain st there is only 1 function from every equivalence class of functions that have the same derivative. But isn't this an invocation of the axiom of choice, which is often not assumed in mathematical discourse? If I do assume that such selections exist, then it still isn't clear how to specify one. If I just forget about trying to specify one, then I think it's easy to prove that all of these "partial inverses" do indeed have the property of linearity, though. (Several of the computer algebra systems I've used implicitly define indefinite integration this way because they only output a single function when you integrate it.)

If I interpret the statement in terms of definite integrals with a variable upper bound, then I guess it's just a weaker statement than the statement in terms of antiderivatives, since the functions that can be expressed in this way are just a subset of the set of all antiderivatives of continuous functions.

So what is usually meant and what is the best way to think when you're solving calculus problems?

Edit: wording

 

[jedishrfu]

I've always thought of indefinite integrals as ones with unspecified integration limits.

Wikipedia gets into it in more detail.

https://en.wikipedia.org/wiki/Antiderivative

 

[Mark44]

I've encountered a few different definitions of "indefinite integral," denoted $\int f(x) \, dx$.

1. any particular antiderivative $F:\mathbb{R} \to \mathbb{R}, F'(x) = f(x)$
2. the set of all antiderivatives $\{F:\mathbb{R} \to \mathbb{R}, F'(x) = f(x)\}$
3. a "canonical" antiderivative
4. any expression of the form $\int_a^x f(x) \, dx$, where $a$ is in the domain of $f$ and $f$ is continuous

For 1, this is part of the Fundamental Theorem of Calculus.
For 2, if F(x) is an antiderivative of f(x), then so too will F(x) + C be, where C is an arbitrary constant. Then $\frac d{dx}(F(x) + C)$ will also equal f(x), as the derivative of the constant is 0.
For 3, I don't remember seeing the term "canonical antiderivative" being mentioned in any of the calculus textbooks I've taught out of during the 20+ years I taught these classes at the college level.
For 4, most textbooks will not write a definite integral such as $\int_a^x f(x) \, dx$, with x appearing as both a limit of integration as well as the "dummy" variable. Instead most textbooks will write such an integral something like this: $\int_a^x f(t) \, dt$, for example.

If I interpret this statement in terms of particular antiderivatives, then it's true for all antiderivatives $\int \left( af + bg \right) \, dx$, $\int f \, dx$, $\int g \, dx$, but only up to a constant, which isn't mentioned.

If a particular textbook doesn't mention that all antiderivatives differ by at most a constant, then they aren't being very precise in their presentation.

If I interpret this statement in terms of "canonical antiderivatives", then I'm not sure it's "even wrong" or "even right" because "canonical antiderivative" isn't fully defined or even shown to exist.

Again, I don't recall seeing this term used in the numerous textbooks I worked out of.

 

[u0x2a]

So the question I'm trying to ask is purely semantic. I'm comfortable solving problems but it's often unclear to me what another person's exact intended meaning is for the term "indefinite integral" or the notation $\int f(x) \, dx$.

For example, here, in this article about indefinite integral misconceptions, the author explicitly defines the indefinite integral as:
$$\int f(x) \, dx = \{F : F' = f(x)\}$$

They then go on to define operations on equivalence classes of the relation $f \sim g \iff f' = g'$ for the purposes of evaluating indefinite integrals.

It seems like there are multiple subtly different ways to think about solving indefinite integrals even though the same notation is often used. For example,
$$\int 2x + 1 \, dx = 2 \int x \, dx + \int \, dx = \frac{2 x^2}{2} + x = x^2 + x$$
If you think in terms of equivalence classes, then the addition is operating on equivalence classes, not individual functions or scalars and the equality is exact. But if you think in terms of individual antiderivatives, then the addition is operating on individual functions and the equality is understood to be equality up to a constant, ie $\exists C \in \mathbb{R} : f = g + C$. They're similar statements but not exactly the same.

I've never seen "canonical antiderivative" in a textbook either but I've heard the idea discussed so I decided to mention it in case anyone had anything illuminating to say about it.

It's interesting that some say "indefinite integral" refers specifically to expressions of the form $\int_a^x f(t) \, dt$. I guess I tend to think of the term as a metonym, where $\int_a^x f(t) \, dt$ is one meaning of the term but the intended meaning is often specifically an operator that can be expressed that way, ie $I: \mathbb{R}^\mathbb{R} \rightarrow \mathcal{P}(\mathbb{R}^\mathbb{R}), I(f) = \{F : F = \int_a^x f(t) \, dt + C\}$, also denoted $\int f(x) \, dx$.

I appreciate the input. However, it's still not that clear to me which of these ways most other people think when they're solving indefinite integrals.

 

[PeroK]

The indefinite integral is an equivalence class of antiderivatives.

The integral $\int_a^x f(t) dt$ is a function of $x$. Or, a function of $a$ and $x$ if you prefer.

 

[jedishrfu]

Try not to overthink this. As has been mentioned, an indefinite integral represents the whole class of integral solutions. It's like integrating up to a point and then waiting for someone to tell you what the limits are before you can find a definite answer.

Math handbooks include tables of indefinite integrals so that busy engineers and scientists can look up an integral rather than go through the steps of solving, because, as we know, there are some really tough nuts to crack. They then apply it to the problem at hand and provide the needed limits to get their answer.

 

[mathwonk]

Exactly for the reason you point out, one should not, in my opinion, ever assume there is a universal definition for any mathematical term. Always consult the specific book you are reading for the definition that will be used ins that book. Even then, some authors are careless and give ambiguous explanations of their language.

The various meanings and their relation, and possible confusion, and possible ambiguity, are discussed in detail in Courant, vol.1, pages 110-116.

 

[wrobel]

I still do not understand what's wrong with the standard
$$\int f(x)dx:=\{F\in C^1(a,b)\mid F'(x)=f(x),\quad x\in(a,b)\},$$
$$\quad f\in C(a,b).$$

 

[wrobel]

It's interesting that some say "indefinite integral" refers specifically to expressions of the form $\int_a^x f(t) \, dt$. I guess I tend to think of the term as a metonym, where $\int_a^x f(t) \, dt$ is one meaning of the term but the intended meaning is often specifically an operator that can be expressed that way, ie $I: \mathbb{R}^\mathbb{R} \rightarrow \mathcal{P}(\mathbb{R}^\mathbb{R}), I(f) = \{F : F = \int_a^x f(t) \, dt + C\}$, also denoted $\int f(x) \, dx$.

$\mathbb{R}^\mathbb{R}$ is the space of all functions from $\mathbb{R}$ to $\mathbb{R}$.
What do you mean by writing
$$\int_a^xf(t)dt$$ with $f$ non- measurable
?