Talk:Integration by substitution

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[edit]  ?

Horrible explanation, incomprehensible to anyone who wants to know, and worthless without those people who'd want to know (otherwise, and obviously, they wouldn't be here to begin with).

Please be more specific. What do you find unclear or confusing? Daniel Sank 16:15, 22 September 2007 (UTC)

[edit] conditions on f

In most calculus books f is required to be continuous (not just integratable), mainly because
their proofs are based on the fundamental theorem of calculus which requires continuity.
See also:
http://eom.springer.de/I/i051740.htm
http://mathworld.wolfram.com/ChangeofVariablesTheorem.html

However i've seen abstract generalization (based on lesbegue integrals) that seems to skip the contintuity condition for f (i do not fully understand the terms involved):
http://planetmath.org/encyclopedia/ChangeOfVariablesFormula.html


It would be nice if somebody knowledgeable could confirm/comment this and modify the article (there should be at least a note why or under which exact condition continuity can be dropped for f).

[edit] which phi?

I replaced φ with Φ, since the latex \phi looks like that one. This may however differ depending on which font you are using...

I don't think thats strictly right. Φ is capital phi, φ is not italicized phi, and φ is italicized phi, which I think is the strictly correct one. I'll leave it up to you to revert if you think it should be done. PAR 20:13, 22 Mar 2005 (UTC)

φ and φ are alternate lowercase forms of phi (uppercase Φ). Both are accessible through LaTeX, via \varphi and \phi, respectively. —Caesura(t) 15:19, 16 October 2005 (UTC)

Fine article. To those responsible: thanks. --Christofurio 17:20, Apr 23, 2005 (UTC)

Small nitpick: Why phi? Can't we use g(x)? Kareeser|Talk! 05:53, 29 June 2006 (UTC)
I've changed the usage of φ(t) (or \varphi(t)) to g(t) in the intro. \phi(\varphi) or whatever is still used later on in another section, but I'll leave that to be edited by whomever. -Matt 17:00, 11 November 2007 (UTC)

I agree with Karesser on this. For most people who will be using the article phi just adds confusion. Is there any reason why it should be phi, because all the calculus books I have seen use g(x). Bizzako 21:54, 16 December 2006 (UTC)

[edit] Requested move

speedy move according to WP standards for upper/lower case in article names. --Trovatore 02:44, 7 December 2005 (UTC)

[edit] Voting

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[edit] Discussion

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[edit] Move issues

I moved this article back from Integration by Substitution. The reasons are three:

  1. The page was moved against Wikipedia conventions for capitals; it should have been integration by substitution
  2. The double redirects were not fixed.
  3. I don't quite see any discussion about why the move would be necessary.

So, if it is decided to move this page indeed to integration by substitution, I can help (you would need an administrator to delete the redirect at integration by substitution first). But I would like to see if people agree on that, and somebody's got to promise to fix the redirects afterwards. Oleg Alexandrov (talk) 03:45, 7 December 2005 (UTC)

Maybe I rushed things a bit with the "requested move" process. I don't have any strong preference for Integration by substitution over Substitution rule; I just tried to move Integration by Substitution to Integration by substitution to comply with the capitalization convention, and then nominated the article when I found out I couldn't just move it. I suppose I think Integration by substitution is just epsilon better than Substitution rule, but I'd be happy to have the nomination quashed and things left as they are now if that's simpler. --Trovatore 04:08, 7 December 2005 (UTC)
Let us see what Atraxani says, the user who did the move (I wrote that user a message). All that really matters is that whoever wants to do the move should fix the redirects, to avoid redirects to redirects. Oleg Alexandrov (talk) 04:13, 7 December 2005 (UTC)
OK, I did the move and fixed the redirects myself. I guess that's the best thing to do. A message to Atraxani though, please do use more care when moving pages. Oleg Alexandrov (talk) 18:11, 9 December 2005 (UTC)

[edit] Multi value thing

If one is required to substitute the variable x in the original integral to something like u^2-u-2, what do you do with the limits? There are obivously two solution of u for each x, so how can one tell which one to use? —Preceding unsigned comment added by 211.31.14.149 (talk • contribs) 23:21, 9 December 2005

You choose a part of the function to use (by imposing constraints) so that u is a (single-valued) function of x. And which one you choose will affect which derivative you get for du/dx. It may be that there are multiple valid choices, depending on your situation; but they should all give the same result. --Spoon! 23:06, 11 September 2007 (UTC)

[edit] integrability of phi

Is it necessary to suppose integrability of (phi)'? Isn't it guaranteed by its assumption of continuity?

[edit] The Calculus Tempate For This Page

It seems that this is the only page that uses the Calculus Template whose link does not become bold and unlinkable when on the page. I don't know how to fix this. Ryulong 20:25, 6 February 2006 (UTC)

That is because the link in the Calculus template used to point to substitution rule. I changed it to point directly to this page, integration by substitution, and now the link does become bold. -- Jitse Niesen (talk) 22:06, 6 February 2006 (UTC)

[edit] Expansion of Substitution rule for multiple variables

It would be great if someone expanded the Substitution rule for multiple variables section with examples and further explanations. Thanks, --Abdull 14:20, 25 May 2006 (UTC)

Yes. Here is something that is rather subtle but bothers people as I know from experience. When the number of dimensions is 1, the multi-variable formula should match the one-variable formula. However, the multi-variable formula has that absolute value in there. In the one-variable formula we have φ'(t) not |φ'(t)|. Explaining why this is not actually a contradiction is an expository challenge. McKay 02:10, 6 March 2007 (UTC)
This is because traditionally when we do a one-dimensional definite integral, we specify a lower bound and an upper bound. When we do the substitution, we apply the function separately onto the bounds. If the substituting function is a decreasing function, then after substitution the lower bound will be greater than the upper bound; and the derivative of the substituting function will be negative. These two effects cancel each other out, as "swapping" the bounds introduces another negative sign to the integral, canceling the negative from the derivative.
However, if you look at how the formula of the multi-variable integral is written, it considers integrals over a set, without considering a specific orientation. And after the substitution, you see that the second integral is over the set which is the image of the function over the original set. If the function has a negative Jacobian determinant, then that means somehow the function "flips" the geometric orientation of the variables. But since we are just integrating sets without considering the geometric orientation, we cannot cancel out that negative. So we take the absolute value. --Spoon! 23:02, 11 September 2007 (UTC)

[edit] It is an unnecessary condition - nonvanishing derivative of g

I think we should remove this unnecessary condition on g : g'(t) \ne 0 \; \forall t \in [a,b] --Novwik (talk) 18:36, 12 January 2008 (UTC)

[edit] Substitution theorem formulated directly for indefinite integral

I have just read a variant of the theorem which is formulated directly for indefinite integral.[1] I think it would fit in the article well. But still, I do not dare to write it to the article, because my knowledge in the topic lacks both the overview and the details.

Instead, I try to summarize the theorem here, together with some remarks, applications and examples. I do not include it here in the talk page literally, because it is too long, thus it may act as distracting the talk page.