Talk:Antiderivative

message: Why isn't this just merged with Integral?--Siva 20:43, 29 Aug 2004 (UTC)
replies: text: Simply put, because antiderivatives and integrals are not the same thing.  I think they deserve separate articles, although certainly a discussion of how they are related belongs in each article (which there currently is).  Remember that it is only by the deep result of the Fundamental theorem of calculus that the seemingly otherwise unrelated concepts of integration and antidifferentiation end up being connected.  and for certain classes of functions, this connection can't be made, because some functions have integrals, but not antiderivatives (within the appropriate domain).  So the connection given by the Fundamental Theorem is deep and important, but it does not mean that the two ideas are exactly equivalent. -lethe talk 01:04, Aug 30, 2004 (UTC)
When is the antiderivative not equal to the integral? What kind of function would have an integral but no antiderivative?--Siva 18:22, 30 Aug 2004 (UTC)

text: For example, the Cantor function. -lethe talk
Also, any constant function f(x) = k. The integral denotes the area underneath the functionbetween bounds. Integrals are inverse differentiation. f(x) = 5 differentiates to f'(x) = 0, but the area beneath f(x) is at no point zero except at x = 0 itself. Another example includes hyporbolas. Integrals crossing any hyperbola's poles will always be positive or negative infinity, as is the definition of a pole. But this is not true of it's antidervite, which is simply a function for which the original hyperbola represents the derivitive.He Who Is 20:34, 28 April 2006 (UTC)

text: That is incorrect. Constant functions have both integrals and antiderivatives, and the two are the same (please read the article, as well as the one on the Fundamental theorem of calculus before commenting). For any continuous function the notions of integral and antiderivative coincide; that's one reason why the two are often conflated. However, the fundamental theorem of calculus does not apply to all functions; to find counterexamples you need to look at discontinuous integrands, e.g. the Cantor function cannot be obtained by integrating its own derivative -- which exists almost everywhere (yet not all discontinuous integrands are counterexamples). FilipeS (talk) 11:32, 15 May 2012 (UTC)
thread_title: Merging with integral?

message: I would not be so sure about a merger. It looks to me that integral is about definite integrals, and antiderivative is the operation inverse to differentiation, meaning input is a function, and output is a function. These are two different things. They are connected through the fundamental theorem of calculus, but this does not mean that they are the same. 
Also note that in more than one variable, integral and antiderivative are even more distinct. As such, I believe two articles are needed, as there are two different concepts, but of course something needs to be said about how they relate. Oleg Alexandrov (talk) 19:30, 4 November 2005 (UTC)
replies: text: Of course it should not be merged.  I agree with Oleg's reasons, and I stand by the reasons I stated over a year ago above -lethe talk 00:18, 5 November 2005 (UTC)
replies: text:  Yes, yes, I forgot to give you credit. :) You indeed wrote about it above, and I used some of that in my argument (but some of my argument was original, even though very similar to what you stated earlier :) Oleg Alexandrov (talk) 01:41, 5 November 2005 (UTC)
replies: text: I think I saw you arguing somewhere that holomorphic and analytic should not be merged.  This is pretty much the same thing (in my opinion).  Just because there's a theorem that says some things are equal, does not mean that the things are the same. -lethe talk 09:12, 5 November 2005 (UTC)

text: The section I added on antiderivatives of non-continuous functions should provide a pretty strong rebuttal to the merger idea Fiedorow 04:12, 11 December 2005 (UTC)
replies: text: Antidifferentiation is the "inverse" of differentiation.  Integrals are entirely different objects. Tomyumgoong 04:05, 29 April 2006 (UTC)

text: Yes, it would be very helpful if the article explain what is the difference between integral and antiderivative.  — Preceding unsigned comment added by 178.42.152.68 (talk) 11:19, 30 March 2013 (UTC)
replies: text: There is no difference. Some authors of calculus books use the expression “indefinite integral”, and others use “antiderivative”, but few use both because the two are synonymous. (However, note that the definite integral is not the same as the indefinite integral/antiderivative.) Dolphin (t) 07:10, 28 December 2021 (UTC)
thread_title:  Merging with [[integral]]? 

message: Oleg would like nicer pictures for Figures 1 and 2 illustrating Example 4. These pictures are given by parametric equations 
:t\mapsto\left(\sum_{n=1}^8\frac{(t-\cos(n))^{1/3}}{2^n},
\frac{1}{\sum_{n=1}^8\frac{(t-\cos(n))^{-2/3}}{3\cdot 2^n}}\right)
:t\mapsto\left(\sum_{n=1}^8\frac{(t-\cos(n))^{1/3}}{2^n},t\right)
respectively, where -1\le t\le and where
:\frac{1}{\sum_{n=1}^8\frac{(t-\cos(n))^{-2/3}}{3\cdot 2^n}}
is assigned the value zero if t=\cos(1),\cos(2),\dots\cos(8). Perhaps someone clever enough with something like gnuplot or Adobe Illustrator or something could produce better pictures. [Note: for the actual Example 4, the upper summation limit of 8 should be replaced by \infty, but that is impossible to plot.]
Also as a matter of curiosity, what is the value of the Lebesgue integral of Example 4?Fiedorow 17:46, 11 December 2005 (UTC)
replies: text: To answer my own question, it seems that Fatou's lemma implies that the Lebesgue integral of Example 4 satisfies the fundamental theorem of calculus.Fiedorow 01:03, 12 December 2005 (UTC)

text: OK, I've now uploaded new versions of the pictures, created within Maple.Fiedorow 17:02, 13 December 2005 (UTC)
replies: text:  Thanks, looks good! The lesson I learn again and again is that it is more productive to bug others than do the work myself. At least, it worked in this case. :)  Thanks a lot! Oleg Alexandrov (talk) 20:23, 13 December 2005 (UTC)
thread_title: Picture business

message: Do you know any computer program to perform these operations? Thank you. --User:Oculto 11 Feb 2006
: Mathematica and Maple (the computer algebra system) can do that. Oleg Alexandrov (talk) 23:42, 11 February 2006 (UTC)
replies: text: Or Maxima, if you prefer free software. See also Comparison of computer algebra systems. A web-based  solution: Wolfram Integrator, based on webMathematica. Torzsmokus 22:23, 21 March 2007 (UTC)
thread_title:  Computer program to take the antiderivative 

message: Shouldn't this page really be titled "primitive" or "primitive function"?  This is the standard term among mathematicians.  It makes sense.  You have a primitive, and its derivative.  Anti-derivative is mostly a term invented by undergraduate textbooks, and seems to me to be a bit unwieldy. Grokmoo 04:25, 19 February 2006 (UTC)
replies: text: Well, I guess undergrad students are a bigger audience here than PhD's. :) I learned this in high school in Europe, where it was called "privitive", but in US I have heard only about "antiderivative". :) Oleg Alexandrov (talk) 06:08, 19 February 2006 (UTC)
replies: text: Yes anti-derivative is a misnomer, anti means contrary not reverse. Primitive is the reverse of derivative. It is sad that in the USA they tend to complicate logical intelligibility by coming up with this type of nomenclature. I support the proposal that the title of primitive should be used.  79.168.11.181 (talk) 17:37, 11 April 2012 (UTC)
thread_title:  Rename to primitive? 

message: From the "Uses and properties" section: 
:"It is critical to remember that an integral is not the same, in general, as the means for evaluating it; and the function that an integral implies stands apart from that means - in the case of single-variable integrals, from antiderivatives."
This sentence is absolutely cryptic. You can understand it only if you know everything a priori (and in this case you don't need to read it)!
I removed it. If it is truly critical to remember, and you want to insert it back, then in my opinion you need 
* to explain its meaning (giving at least an example)
* to explain why it is so important to remember. 
With regards,
Paolo.dL 09:02, 2 August 2007 (UTC)
thread_title:  Crucial but absolutely cryptic sentence 

message: Here's the definition of "indefinite integral" given in the introduction of the article:
: "In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F..."
And here's the definition of "indefinite integral" given in the section Uses and properties:
: "Because of this, the set of all antiderivatives of a given function f is sometimes called the general integral or indefinite integral of f"
Can you see the inconsistency between the two definitions? I believe that the second is not correct. An indefinite integral is not "all the antiderivatives of f" (an infinite set of functions), but just "any antiderivative of f", i.e. "a single antiderivative" arbitrarily selected from among infinite possible ones. This is because c, the integration constant, is not a set, but just a single constant arbitrarily selected from among a set.
Think of the nonunique solution of a square root, another example of quasi-inverse operation: 
: square \ root(a^2) = \pm \sqrt {a^2} = \pm a. 
:(I couldn't find a standard symbol for this operation; sqrt(x) or sqr(x) are conventionally used to indicate \sqrt x, the principal square root of x)
The symbol ± means either "+ or -". It does not mean both "+ and -". Therefore, it would be incorrect to write:
: square \ root(a^2) = \{ +a, -a \} \,
where {+a, -a} is the set containing +a and -a. In symbols:
: square \ root(a^2) \neq \{ +a, -a \} = \begin{cases} +a \\ -a \end{cases} 
Paolo.dL 23:41, 29 July 2007 (UTC)
thread_title:  Introduction 

message: So, there exist different approaches in the literature. However:
* Unwanted ambiguity. Wikipedia was the only page where the integral was defined in two different ways. In a single article, you must either use a single definition, or explicitly state that different definitions exist in the literature.
* Ambiguity of the word "arbitrary". This statement will prove to be useful later. The word "arbitrary" in the expression "arbitrary constant of integration" is somewhat ambiguous. It may be used to mean:
:# "arbitrarily selectable" from among infinitely many values (hence still indeterminate, until selection is accomplished), or
:# "arbitrarily selected" (hence determined through a process of "arbitrary selection of one value C from among all possible values \mathbb{R} or \mathbb{C}").
* Does the integral symbol always mean integral? As far as I know, everybody writes indefinite integral examples using C (or c) as constant term, and they always call C the "arbitrary constant of integration". This is a widely accepted standard terminology, which I have seen repeated everywhere. Is there anybody who calls C the "infinite set of integration constants"? 
: If the indefinite ingegral were defined as a set, and the symbol \int \, meant "indefinite integral" (rather than a single antiderivative), then the definition formula would contain a set in the second member. For instance, (I am not sure if this notation is correct): 
:: \int f(x)\, dx = \{ F(x) + c: c \in \mathbb{R} \} 
:: 
: or, simplifying (I don't know if this notation is allowed):
:: \int f(x)\, dx = F(x) + \mathbb{R}
: where F(x) + \mathbb{R} is used as if it were the sum of a scalar plus a matrix.
: Since I have never seen an indefinite integral written in a similar way, I hypothesize that those who maintain that an indefinite integral is a set might use the symbol \int \, to indicate an antiderivative, rather than an indefinite integral. In this case the indefinite integral would remain an operation without a symbol, as the square root above. Is my hypothesis correct?...
* Quasi-inverses (nonunique inverses). The indefinite integral may be regarded as indeterminate. Partially determinate, but all the same indeterminate. It determines a set of possible results, i.e. a nonunique result. This is also true for the square root. Both the indefinite integral and the square root can be reagarded as nonunique inverses or "quasi-inverses". Now, what do we do with nonunique results? We can either arbitrarily select one of them or just take them all, as a set. And I am not certain about what is the best option. But even if we arbitrarily select one of the possible results, they are still nonunique and the integral is still indeterminate.
* Pseudo-inverses. I give to the term "pseudo-inverse" a different meaning. In my opinion, pseudo-inverse is an operation which gives a unique result, such as the Moore-Penrose pseudo-inverse, or the principal square root, or "the simplest" antiderivative (when defined), or a definite integral with variable upper limit. Therefore, a pseudo-inverse is pseudo because it uses as input some additional information which is not typically available to an inverse function (e.g. the assumption of orthogonality, or an arbitrarily selected sign for a square root, or an arbitrarily selected value for the constant of integration or for the lower limit of integration). 
* 2nd theorem of calculus. When the integral is defined as an arbitrarily selected antiderivative, the second fundamental theorem of calculus is written and explained as follows: 
: \int_a^b f(x)\,dx = F(b) - F(a),  where F = indefinite integral = single antiderivative
: The statement F = indefinite integral wouldn't work if the indefinite integral were a set! F(b) - F(a) would be the subtraction of two sets, and (if we perform an elementwise subtraction) would yield an infinite set of identical numbers. There's a possible approach to solve this problem, which appears not very satisfactory to me: 
: \int_a^b f(x)\,dx = F(b) - F(a),  where F = single antiderivative ≠ indefinite integral (indefinite integral = set of antiderivatives)
: Here you define the indefinite integral as a set, but then you declare that F is just an element of that set. In this case the theorem would not relate the definite integral with the indefinite integral, but just with one element of the indefinite integral set. The indefinite integral would not explicitly appear in the theorem. That's a strong limitation of this solution, in my opinion. Why should we define the indefinite integral as an object which does not appear in the theorem?...
* Right-invertible. Correct me if I am wrong: if the result of an indefinite integral were a set, then the derivative of that set would be a set of infinitely many identical functions. This would make no sense. The indefinite integral (as well as the definite integral with variable upper limit) should be reversible by a derivative:
:* An indefinite integral cannot fully reverse a derivative (i.e. it is not left-invertible), but we are not masochistic: 
:* why shouldn't it at least be reversible by a derivative? (i.e. right-invertible).
:(see the two conditions required to be a complete inverse, in the article inverse function, that I just finished to edit yesterday; see also the first part of the fundamental theorem of calculus).
:Again, the example about square root can help. A square root is reversible by a square, isn't it? If a2 = b:
: square \ root(a^2) = \begin{cases} +a \\ -a \end{cases}  (not left-invertible; nonunique solution)
: square \ root(b)^2 = b  (right-invertible; unique solution) 
:Am I right, or even in this case different definitions of the square root can be found in the literature?
* Tentative conclusion. Based on what little I know, I conclude that defining the indefinite integral as a set is possible, but it entails an appearently unreasonable entanglement. If there is a good reason to choose this approach, I cannot see it. If we want to describe this approach on the article, the rationale should be clearly explained.
* Reliable sources. By the way, are we sure that the Encyclopaedia of Mathematics (2002) , and Calculus for Dummies p. 236:  or Wikipedia or MathWorld or PlanetMath are reliable sources? Shouldn't we trust only university textbooks dealing with the specific topic? Encyclopedias and divulgative books are not primary sources.
Paolo.dL 10:53, 5 August 2007 (UTC)
thread_title:  Analysis and classification of the different approaches 

message: There has been vandalism here lately.  Even if the person thinks that his/her point is true, he/she needs to realize that going about it that way is unencyclopedic and can result in his/her IP address being blocked.  All controversial claims need to be properly sourced.  Here are the 2 vandalistic revisions:
http://en.wikipedia.org/w/index.php?title=Antiderivative&oldid=324283950
http://en.wikipedia.org/w/index.php?title=Antiderivative&oldid=324270368
The IP address is 81.141.239.187
This is to bring this issue into the open. Cornince (talk) 15:13, 6 November 2009 (UTC)
thread_title:  Vandalism 

message: I did not know this term dispite to be a mathematicain, sorry. :) It sounds really new fashion -- (indefinite) integral is the common wording since more than 100 years, IMHO.
Or is this an invention for/of Wikipedia?
I think such a recent naming should be simple to be traced to its origin. :)
Thanks in advance for clearification -- probably also the reader of the article likes to know this. Achim1999 (talk) 20:58, 13 October 2012 (UTC)
thread_title:  Who coined this term "antiderivation"? 

message: The illustration (at the upper right of the article) does more harm than anything else.
It has no clear relation to anything in the article.  
Its caption uses the word "solution" but says nothing about what the problem is that it is a solution to.
Some illustration + caption would be a good idea, but not this one.  I vote to remove it.Daqu (talk) 03:48, 2 March 2015 (UTC)
thread_title:  Illustration should be removed 

message: Hey, DVdm, what's the matter with you? There is really no need to undo my math formatting, is there? It took me quite a while to clean up that article ...! So please stop reverting unless you have a good reason.--Neufund (talk) 17:38, 4 September 2018 (UTC)
replies: text: Regarding , see MOS:FORMULA: "...the community of mathematical editors of English Wikipedia currently has no consensus about preferred formatting...", so this can be discussed here. Don't worry about your Sisyphean work: if others agree that we should indeed change the article, then we can simply undo the undo. - DVdm (talk) 17:45, 4 September 2018 (UTC)

text: I'm going to remain a bit agnostic on which formatting method to use for a minute, but I did notice a few specific problems with your changes that I want to point out.  In running text, it's usually better to write simple fractions with a slash: x^3/3 instead of \frac{x^3}{3}.  Sometimes it can't be helped, but this helps prevent widely varying line widths.  Also, things like \exp don't take any arguments (which isn't technically the right way to describe it, but whatever).  So you should have \exp x^2 rather than \exp{x^2}.  –Deacon Vorbis (carbon • videos) 17:52, 4 September 2018 (UTC)
replies: text: (after ec) As I already tried to explain to DVdm on my talk page: Please consider the fact that there already was some math inline formatting (e. g. at the beginning) before. So I just tried to harmonize the formatting. Otherwise, one would have to remove all of the math inline formatting. So, you can go ahead if you like, but what we certainly don't want is a mishmash of both non-math and math formatting of mathematical expressions here, do we?!--Neufund (talk) 17:55, 4 September 2018 (UTC)

text: PS: As for your particular points of criticism: Feel free to lend a hand ... 😉--Neufund (talk) 17:57, 4 September 2018 (UTC)
replies: text: I'm all for harmonizing within one article—not within the entire Wikipedia of course . So I'd prefer converting inline math to standard inline, but in this case, some rendered math is a bit hard to do inline. That's probably the reason of the mix here. - DVdm (talk) 18:11, 4 September 2018 (UTC)
replies: text: ... which is why I opted for a consistent use of inline math ... 😊 It's certainly not wrong, is it?--Neufund (talk) 18:14, 4 September 2018 (UTC)
replies: text:   — Oops. Sorry. My mistake. Inline meant to me just italics, like a2 − b2. Not like  () or  a^2-b^2 (). Feel Absolutely Free to restore your work, taking Deacon Vorbis' suggestions into account. Sorry again... - DVdm (talk) 18:28, 4 September 2018 (UTC)
thread_title:  Formatting 

message: recently removed all external links on this page and  replaced them. I agree with their removal. External links are to satisfy certain narrowly defined objectives and should not be used as a catchall for websites concerned with the article's topic. More specifically, I would remove any on-line calculator on the grounds that it is not a reliable source (the only exception to this is if a calculator has been vetted by a reliable source in a citable manner). The other links should be removed in accordance with WP:ELNO #1. --Bill Cherowitzo  (talk) 18:13, 29 September 2018 (UTC)
replies: text: I more or less agree. My replacement by working links with better structure is only a temporary fix. As final fix I suggest to:
* Remove all external links explaining antiderivatives
* Merge any useful extra info from the external links in this article. (This will probably be barely anything)
* Mention in the article it self that there are sites that can, with or without explanation, integrate functions. This could be cited with one or more of the "tool-sites" in the external links --Garo (talk) 00:03, 30 September 2018 (UTC)
replies: text: I disagree with a complete removal. If I recall correctly, there is already precedence of dealing more accurate and sensibly with external links, that offer valuable information (graphic, algorithmic, animated, ...), than is done with this here non-selective purging, or at other sites with AfDs, and analogously. WP is not bound to obey always to the most rigorous, but may act against "fiat iustitia, pereat mundus". Purgy (talk) 09:07, 30 September 2018 (UTC)

text: (Oops, I got distracted and forgot to reply here).  Well, there's already a brief mention that computer algebra systems can do symbolic integration; whether or not one has a web interface doesn't seem all that important.  But even if any mention is made, any link should only be to the site's Wikipedia article, if it exists.  I think Wolfram Alpha is the only one that qualifies (the current link to the Integrator isn't even used by them as a standalone thing anymore; it just redirects to Alpha).  It still wouldn't qualify as an external link though (it  at the Wolfram Alpha article itself, but not here).  –Deacon Vorbis (carbon • videos) 15:34, 30 September 2018 (UTC)
thread_title:  External links 

message: The 4th equation in section "Uses & properties" gives "F(x) = \int_0^x f(x)\,\mathrm dx" which isn't valid already for the example given just before, f(x) = 1/x². I think the lower limit should definitely be replaced by "a", for example, and "a ∈ Df" be appended to the line (although that remains a simplification for the case of disconnected domain). — MFH:Talk 16:08, 13 April 2023 (UTC)
thread_title:  particular primitive for continuous functions 

message: The phrase
:...many functions whose antiderivatives ... cannot be expressed (...). Examples of these are
should be followed by the functions, but it is followed by the primitives. — MFH:Talk 16:18, 13 April 2023 (UTC)
thread_title:  "Examples of functions..." gives primitives instead of the functions 

message: In the physics example I think the + C should be part of the v, in other words it should not be there. That constant is part of the velocity not separate from it. 2.36.106.75 (talk) 17:57, 13 April 2024 (UTC)
replies: text: I disagree. The term “+ C” is the constant of integration. Dolphin (t) 22:32, 13 April 2024 (UTC)
thread_title:  Physics example