Talk:Antiderivative

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[edit] Merging with integral?

Why isn't this just merged with Integral?--Siva 20:43, 29 Aug 2004 (UTC)

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)

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)

[edit] Merging with integral?

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)

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)
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)
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)
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)
Antidifferentiation is the "inverse" of differentiation. Integrals are entirely different objects. Tomyumgoong 04:05, 29 April 2006 (UTC)

[edit] Picture business

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)

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)
OK, I've now uploaded new versions of the pictures, created within Maple.Fiedorow 17:02, 13 December 2005 (UTC)
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)

[edit] Computer program to take the antiderivative

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)
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)

[edit] Rename to primitive?

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)

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)

[edit] what's with the revertion?

Why is Oleg Alexandrov reverting my addition of alternate names of antiderivative in the intro? Moreover, the use of 3 'or's in sucession is very ugly to read. Loom91 10:59, 14 August 2006 (UTC)

Here is the diff. One should not insert links in definitions, so integral at the beginning is inappropriate.
The connection between derivatives and integrals is in the next sentence in the intro, there is no need to put it in the definition. Oleg Alexandrov (talk) 16:07, 14 August 2006 (UTC)
The 'or's are not nice, but there is no need for integral as well as indefinite integral, and Oleg is right about the link. JPD (talk) 16:23, 14 August 2006 (UTC)
I'd agree with Oleg. Antiderivatives are not integrals. The connection is explained just a short breath later. I'd say keep "integral" and "indefinite integral" out of the first sentence.
On a different note, I think I'd take out the sentence, "Finding an expression for an antiderivative is harder than calculating a derivative, and may not always be possible." It kinda clutters the intro, and isn't quite correct -- or rather isn't quite precisely worded. You might say, "Finding a closed-form expression for an antiderivative for a composition of elementary functions is harder than calculating a derivative of such an expression, and may not always be possible," but that's getting kinda wordy.
My two cents, Lunch 17:27, 14 August 2006 (UTC)
I agree with Oleg. Keep "indefinite integral" which is the same thing, and take out "integral" which is not the same thing. If people are concerned with two "or"s (which isn't really such a big deal) use "antiderivative, primitive, or indefinite integral". VectorPosse 21:37, 14 August 2006 (UTC)

[edit] Renaming to Primitive

For the sake of pluralism I think primitive should be added to the title, as i had never heard of an antiderivative until i tried to look up primitivation... In europe the word antiderivative is used only in england... In french and german for example they say "primitive", in portuguese and spanish "primitiva" and little or no references to antiderivative can be found in textbooks, undergraduate or not :D Sebastiao 14:37, 5 December 2006 (UTC)

Since it is called an antiderivative in US and England, and probably in other countries whose main language is English, I guess it is better if it stays the way it is now. The name "primitive" is mentioned in the first article sentence though. Oleg Alexandrov (talk) 15:47, 5 December 2006 (UTC)
In India we usually use indefinite integral, though primitive is also sometimes used. I don't remember seeing antiderivative. Are you sure England primarily uses antiderivative? Loom91 16:27, 5 December 2006 (UTC)
Also, the disambiguation page for the word "primitive" refers to this page (although it's a bit buried down the page). I might comment that there are all sorts of math terms that translate into other languages in confusing ways. For example, the words for "manifold" in Spanish and French (and probably all Romance languages) are cognates of the English word "variety", but we can't refer to manifolds as varieties for the sake of people from other countries looking for an article on manifolds. A variety and a manifold are, in English, two very different things. (At least the word "primitive" also has the meaning of "antiderivative" so it's not a problem in this case.) VectorPosse 16:50, 5 December 2006 (UTC)
FYI: As far as I know, "indefinite integral" is used universally in high-school and undergrad levels in England. At least it was in the 80s and 90s when I was a student there - I had never heard of "antiderivative" before I got to this page. —The preceding unsigned comment was added by Dave w74 (talk • contribs) 03:11, 21 February 2007 (UTC).
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