What Is the Correct Evaluation of the Indefinite Integral of Zero?

[PFuser1232]

Argument A

$∫ 0 dx = 0x + C = C$

Argument B

$∫ 0 dx = ∫ (0)(1) dx = 0 ∫ 1 dx = 0(x+C) = 0$

Discuss.

 

[Mark44]

The "Discuss" part makes me think this is a homework problem.

 

[PFuser1232]

The "Discuss" part makes me think this is a homework problem.

It's not, I just wanted to know which argument is right and why.

 

[Mark44]

Argument B is silly, as it factors 0 into 0 times 1, and then moves the 0 outside the integral.

 

[PFuser1232]

Argument B is silly, as it factors 0 into 0 times 1, and then moves the 0 outside the integral.

What about the rule $∫a f(x) dx = a ∫ f(x) dx$?

 

[pwsnafu]

The indefinite integral does not calculate a function. It calculates an equivalence class of function. All constant functions are in the same equivalence class.

Further, $\int 0 \, dx$ being constant is wrong. If for example the domain is $\mathbb{R}\setminus\{0\}$ then you would have two constants, not necessarily equal.

 

[Mark44]

Argument A

$∫ 0 dx = 0x + C = C$

There's no need to write 0x in the above.
$\int 0 dx = C$

Argument B

$∫ 0 dx = ∫ (0)(1) dx = 0 ∫ 1 dx = 0(x+C) = 0$

$\int 0 dx = 0\int dx = 0x + C = C$
You don't get rid of the arbitrary constant as you showed above.

Since d/dx(C) = 0, then any antiderivative of 0 is C, for some arbitrary constant.

Discuss.

 

[Mark44]

Further, $\int 0 \, dx$ being constant is wrong. If for example the domain is $\mathbb{R}\setminus\{0\}$ then you would have two constants, not necessarily equal.

Can you elaborate on this? I'm not following you. How does the domain enter into this problem, given that we're working with indefinite integrals?

 

[PeroK]

Argument A

$∫ 0 dx = 0x + C = C$

Argument B

$∫ 0 dx = ∫ (0)(1) dx = 0 ∫ 1 dx = 0(x+C) = 0$

Discuss.

For what it's worth, I think this is an interesting question. What do you think? A or B?

Hint: one answer is clearly wrong, but it's not so easy to explain exactly why.

 

[pwsnafu]

Can you elaborate on this? I'm not following you. How does the domain enter into this problem, given that we're working with indefinite integrals?

This is not easy to see when you have $f(x) = 0$, so the example I give my students is
$f : \mathbb{R}\setminus\{0\} \rightarrow \mathbb{R}$ where $f(x) = -x^{-2}$ and
$g: (0, \infty) \rightarrow \mathbb{R}$ where $g(x) = -x^{-2}$.
Then the function
$h(x) = x^{-1} + 1$ for $x > 0$ and $h(x) = x^{-1} - 1$ for $x < 0$ is an antiderivative of f. But it isn't expressible as $F(x) + C$ where C is an "arbitrary constant". The "constant" changes as you pass over the "hole". On the other hand, h isn't an antiderivative of g (because the domains don't match), but the restriction of h to $(0,\infty)$ is an antiderivative of g and expressible in the form $G(x) + C$.

The difference lies in the number of connected components of the domain. The former has two so there are two constants of integration, one for x<0 and another for x>0. The latter only has one connected component, and there is one constant of integration. In general $\int f(x) \, dx = F(x) + H(x)$ where $F' = f$ and $H$ is constant on individual connected components of the domain of f.*

Writing $f(x) = 0$ is ambiguous. It most likely has domain of all of reals. But it could be any subset of R. OP did not specifiy the domain, hence he cannot make the conclusion that the constant of integration is indeed a constant value always.

*Edit: I'm ignoring functions like Cantor's functions here.

 

[PFuser1232]

The indefinite integral does not calculate a function. It calculates an equivalence class of function. All constant functions are in the same equivalence class.

Further, $\int 0 \, dx$ being constant is wrong. If for example the domain is $\mathbb{R}\setminus\{0\}$ then you would have two constants, not necessarily equal.

So the indefinite integral is basically a set of functions?
A set of antiderivatives, more specifically.

 

[PFuser1232]

For what it's worth, I think this is an interesting question. What do you think? A or B?

Hint: one answer is clearly wrong, but it's not so easy to explain exactly why.

My gut tells me it's A.

 

[PeroK]

My gut tells me it's A.

Your head should tell you that as well. In general, you have to be careful with the constant of integration. It's an informal way of specifying an equivalence class of functions (as pointed out above). So, that's where the multiplication by 0 breaks down. If you do things more formally, you still have an equivalence class after multiplication by 0 - so you don't get rid of the constant of integration.

And, yes, an indefinite integral is actually a set of functions.

 

[PFuser1232]

Your head should tell you that as well. In general, you have to be careful with the constant of integration. It's an informal way of specifying an equivalence class of functions (as pointed out above). So, that's where the multiplication by 0 breaks down. If you do things more formally, you still have an equivalence class after multiplication by 0 - so you don't get rid of the constant of integration.

And, yes, an indefinite integral is actually a set of functions.

And how exactly do I do things more formally?
Does $∫a f(x) dx = a ∫ f(x) dx$ only apply for nonzero a?

 

[PeroK]

And how exactly do I do things more formally?
Does $∫a f(x) dx = a ∫ f(x) dx$ only apply for nonzero a?

No. You work with equivalence classes of functions, not with a constant of integration.

An analogy would be $0 \cdot x(mod \ n) = 0 (mod \ n) ≠ 0$

 

[PFuser1232]

No. You work with equivalence classes of functions, not with a constant of integration.

An analogy would be $0 \cdot x(mod \ n) = 0 (mod \ n) ≠ 0$

So what you're saying is I should treat the indefinite integral as a set of functions ${F(x) + C | C ∈ ℝ}$, but not as a single function?

 

[Mark44]

So what you're saying is I should treat the indefinite integral as a set of functions ${F(x) + C | C ∈ ℝ}$, but not as a single function?

Yes.

If a function f has two distinct antiderivatives F₁ and F₂ (IOW F₁] = f and F₂] = f) then F₁(x) - F₂(x) ≡ C.

An example in the same vein is this:
$$\int sin(x)cos(x)dx$$
One student uses substitution to evaluate this integral, using u = sin(x), so du = cos(x)dx.
Then integral becomes $\int udu = (1/2)u^2 = (1/2)sin^2(x)$.

Another student also uses substitution, but with u = cos(x), du = -sin(x)dx
With this substitution, the integral becomes $-\int udu = -(1/2)u^2 = -(1/2)cos^2(x)$.

(Notice that both students omitted the constant of integration.)

Here we have two distinct antiderivatives for the same integrand. As it turns out, the two antiderivatives differ by a constant: (1/2)sin²(x) = -(1/2)cos²(x) + 1/2, independent of the value of x.

 

[Stephen Tashi]

What about the rule $∫a f(x) dx = a ∫ f(x) dx$?

As PeroK mentioned, such rules are not about symbols representing unique functions. The symbol $\int f(x)$ does not represent a unique function. Such rules don't work if you interpret the anti-derivative symbol to be a unique function.

For example If suppose we say that $\int x dx$ represents unique function $x^2/2 + 1$.
and the rule you quoted says $\int 2x$ is the unique function $2( x^2/2 + 1) = x^2 + 2$. This would contradict any calculation that said $\int 2 x = x^2$ or $\int 2 x = x^2 + 5$ etc.

 

[mathman]

For indefinite integrals, the arbitrary added constant is just there. All you are doing in B is omitting it.

 

[PFuser1232]

Yes.

If a function f has two distinct antiderivatives F₁ and F₂ (IOW F₁] = f and F₂] = f) then F₁(x) - F₂(x) ≡ C.

An example in the same vein is this:
$$\int sin(x)cos(x)dx$$
One student uses substitution to evaluate this integral, using u = sin(x), so du = cos(x)dx.
Then integral becomes $\int udu = (1/2)u^2 = (1/2)sin^2(x)$.

Another student also uses substitution, but with u = cos(x), du = -sin(x)dx
With this substitution, the integral becomes $-\int udu = -(1/2)u^2 = -(1/2)cos^2(x)$.

(Notice that both students omitted the constant of integration.)

Here we have two distinct antiderivatives for the same integrand. As it turns out, the two antiderivatives differ by a constant: (1/2)sin²(x) = -(1/2)cos²(x) + 1/2, independent of the value of x.

I have always tackled such problems by using two different constants of integration for the two antiderivatives, then relating them according to the two results.

 

[PFuser1232]

As PeroK mentioned, such rules are not about symbols representing unique functions. The symbol $\int f(x)$ does not represent a unique function. Such rules don't work if you interpret the anti-derivative symbol to be a unique function.

For example If suppose we say that $\int x dx$ represents unique function $x^2/2 + 1$.
and the rule you quoted says $\int 2x$ is the unique function $2( x^2/2 + 1) = x^2 + 2$. This would contradict any calculation that said $\int 2 x = x^2$ or $\int 2 x = x^2 + 5$ etc.

So technically, if we're doing a rigorous treatment of indefinite integrals, would the statement $∫ 1/x dx = ln|x| + C$ (for example) be wrong because it would imply that the indefinite integral is a single unique function? Should we draw a line between the often interchangeable terms "Indefinite Integral" and "Antiderivative" by expressing Indefinite Integrals as sets of all possible Antiderivatives? Or are they synonyms?

 

[Stephen Tashi]

it would imply that the indefinite integral is a single unique function?

It doesn't specify a unique function unless you interpret $C$ as symbolizing a unique number.

Should we draw a line between the often interchangeable terms "Indefinite Integral" and "Antiderivative" by expressing Indefinite Integrals as sets of all possible Antiderivatives? Or are they synonyms?

There are areas of mathematics where terminology is imprecise. I think most peopl treat "indefinite integral" as a synonym for "antiderivative". Many people say "the" indefinite integral or "the" antiderivative even though they know the functions are only unique "up to a constant". If we want to study the symbolic calculations of calculus in a rigorous way as a manipulation of strings (the way a symbolic algebra computer program woud do them) then we'd have to use more precise terminology. As I recall, a mathematician named Ritt did such work. We can investigate what terminology he used.

 

[Nick O]

I just skimmed through the answers and didn't see a comment like this. I'm sorry if I missed it and am just repeating what someone else said.

You can safely think of an antiderivative as exactly what the name says: the inverse of a derivative.

d/dx 5 = 0.

So, the antiderivative of 0 dx has to be the same shape as 5, namely, a constant.

Edit:

So what you're saying is I should treat the indefinite integral as a set of functions ${F(x) + C | C ∈ ℝ}$, but not as a single function?

I don't think there is any reason to restrict C to the reals. C could just as easily be 5+3i, because d/dx (5+3i)=0

 

[pwsnafu]

You can safely think of an antiderivative as exactly what the name says: the inverse of a derivative.

d/dx 5 = 0.

So, the antiderivative of 0 dx has to be the same shape as 5, namely, a constant.

The derivative is not injective. It does not have an inverse. It has a right inverses.

I don't think there is any reason to restrict C to the reals. C could just as easily be 5+3i, because d/dx (5+3i)=0

You restrict to the co-domain of the function you are integrating. If the OP chose R, you restrict the constant to R.

 

[PFuser1232]

The derivative is not injective. It does not have an inverse. It has a right inverses.
You restrict to the co-domain of the function you are integrating. If the OP chose R, you restrict the constant to R.

Agreed.
But I am not questioning the validity of the fact that an antiderivative of 0 is just a constant, I was questioning the validity of the constant multiple rule for indefinite integrals.

 

[PFuser1232]

It doesn't specify a unique function unless you interpret $C$ as symbolizing a unique number.

There are areas of mathematics where terminology is imprecise. I think most peopl treat "indefinite integral" as a synonym for "antiderivative". Many people say "the" indefinite integral or "the" antiderivative even though they know the functions are only unique "up to a constant". If we want to study the symbolic calculations of calculus in a rigorous way as a manipulation of strings (the way a symbolic algebra computer program woud do them) then we'd have to use more precise terminology. As I recall, a mathematician named Ritt did such work. We can investigate what terminology he used.

So should I interpret the indefinite integral of a function as a "set", or as a function with a "variable constant of integration"?

 

[PFuser1232]

So should I interpret the indefinite integral of a function as a "set", or as a function with a "variable constant of integration"?

?

 

[Stephen Tashi]

So should I interpret the indefinite integral of a function as a "set", or as a function with a "variable constant of integration"?

What distinction are you making between a "set" and "a function with a variable constant of integration"? From an abstract point of view a an expression for a function with a "variable constant" defines a set of functions.

How you should think about the symbols in the rules of integration depends on your purposes. If you are trying to develop computer programs (like Mathematica) to do symbolic calculations then you may have to get into very technical jargon. Is that what you are trying to do?

 

[PFuser1232]

As PeroK mentioned, such rules are not about symbols representing unique functions. The symbol $\int f(x)$ does not represent a unique function. Such rules don't work if you interpret the anti-derivative symbol to be a unique function.

For example If suppose we say that $\int x dx$ represents unique function $x^2/2 + 1$.
and the rule you quoted says $\int 2x$ is the unique function $2( x^2/2 + 1) = x^2 + 2$. This would contradict any calculation that said $\int 2 x = x^2$ or $\int 2 x = x^2 + 5$ etc.

Well, I could say $\int x dx = \frac{x^2}{2} + C$ where $C$ can take on any value. We can then find $\int 2x dx$ as follows:
$\int 2x dx = 2\int x dx = 2[\frac{x^2}{2} + C] = x^2 + 2C$
Now, $2C$ is a constant just as well as $C$; we might as well call them $C_1$ and $C_2$, doesn't matter. What matters is they're both "constants" of integration (and yes, they can take on many values). There's no problem with this example. 2 is not a problem, 0 is. The product of 0 and anything is 0, how can we then still keep the constant C (or variable) without violating this rule?

 

[vela]

As has been mentioned, $\int f$ is not a single function but an equivalence class of functions. To make this explicit, I'll denote this set as $[\int f]$. If a function F is an element of $[\int f]$, we mean that $F' = f$.

When we say that $\int cf = c \int f$, we mean that the set $[\int cf]$ and the set $c[\int f]$ are equal. The first set is easy enough to understand: if F is in the set $[\int cf]$, then $F'(x) = (cf)(x)$ for all x. But what does $c[\int f]$, a scalar multiplying a set, mean? Your example posits if $F \in c[\int f]$, then $F = c\cdot G$ where $G$ is some element of $[\int f]$. It follows then that $c=0$ implies $F=0$. This leads to the contradiction you noted, so it's not the right interpretation if we want the notation to make sense. If instead we say that if F is an element of $c[\int f]$, we mean that $F'(x) = c\cdot f(x)$ for all x, there's no problem. If $c=0$, we have $F' = 0$, which implies F is constant. Moreover, since $(cf)(x) = c\cdot f(x)$, it's clear the two sets are equal for any value of c.

If you're familiar with solving differential equations, you can look at it this way as well: Differentiating $F = \int cf$ or $F = c \int f$ yields the same differential equation F' = cf. The general solution to this differential equation consists of a homogeneous part, which is the solution to F'=0, plus a particular solution F_p where F'_p = cf. If $c=0$, you lose the particular solution, but you're still left with the homogeneous solution, which is the constant of integration. In other words, the constant $c$ doesn't affect the constant of integration.