Until a few days ago, every calculus textbook I have seen uses $u$ as the default variable for integration by substitution (a.k.a. integration by change of variables). It was brought to my attention recently that there is an exception: the calculus textbook by Hughes-Hallett et al. uses $w$ as its default letter for integration by substitution. The reason behind this is presumably to distinguish between the variables used in integration by substitution and in integration by parts. While that is an admirable idea, in practice I'd never seen anything other than "$u$-substitution" before. (As a graduate student I did teach with a much earlier edition of Hughes-Hallett than the current one but I do not recall the change of variables in the book being $w$, so I don't know if this $w$-substitution was present from the first edition of the book or not.)

Does anyone know calculus books besides Hughes-Hallett that discuss integration by substitution with a default letter other than $u$, or do you teach it with a default letter other than $u$ even if the class textbook uses $u$? (Edit: I do not have in mind trigonometric substitutions, where it is natural to use the new variable $\theta$ rather than $u$ for traditional geometric reasons. That also has a different flavor than what happens with other instances of integration by substitution in a calculus class, since standard substitutions take the form $u = f(x)$ while trig substitution is $x = f(\theta)$, so I am not asking about trigonometric substitution here.)

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    $\begingroup$ When I teach the change of variables, I don't use a default letter. I very intentionally mix it up quite a bit to get the point across that the letter is just a stand-in for something else. I even throw in the occasional $\tau$ or smiley-face, just to keep things interesting. Ditto integration by parts. $\endgroup$
    – Xander Henderson
    Jan 29, 2018 at 2:17
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    $\begingroup$ I'm very thankful that in russian the method is just called "substitution" or "change of variable", and never ever "u-substitution". It sounds very strange to me to focus so much on which letter we use. Especially, since one substitution is never enough, and I usually have to go through half the alphabet $\endgroup$
    – Yuriy S
    Feb 6, 2018 at 13:12
  • $\begingroup$ Trigonometric substitution and the so-called $u$-substitution differ more fundamentally than merely in using different alphabets: the former is an implicit substitution and must be invertible, unlike the latter. $\endgroup$
    – ryang
    Feb 8, 2023 at 18:44
  • $\begingroup$ On a related note, this shorthand declines to explicitly change the variable even whilst essentially performing integration-by-change-of-variable. $\endgroup$
    – ryang
    Feb 9, 2023 at 4:46

1 Answer 1


Small but not cherry-picked sample. Books I have in library.

  1. Granville (40s edition but goes back earlier for first editions): Uses $z$ for the $\tan(z/2)$-substitution and for the $z^n$ substitution. And for the random extra substitutions. But note also discusses integrals of forms like $\sqrt{a^2-u^2}$. In other words, using $u$ instead of $x$ for these tables of random integrals. (So not preserving it for $u\,dv$ parts integration only.)

  2. Thomas Finney, c. 1980. Similar to Granville with using $z$ as preferred new variable and other variable names. Does have one example with a double substitution and in this case, they use $z$ to start and then $\theta$.

  3. Swokowski, early 80s. Uses $\theta$ for most of the random trig substitutions. Uses $u$ for the hyperbolic trig substitutions. Uses $z$ for the $\tan(z/2)$ master-blaster substitution. Uses both $u$ and $z$ for the random other substitutions. Uses both $a^2-u^2$ and $a^2-x^2$ in discussion special forms (i.e., not protecting $u$, but also sometimes just using $x$). The integration by parts is still normal $u\,dv$. Seems to use $z^n$, for that substitution, but not sure if I found that correctly.

Net/net: seems like $z$ and $\theta$ like to makes some appearances also.

  • $\begingroup$ What do you mean by asq, usq, and xsq? Looks like a bad typo. And what's IOW? $\endgroup$
    – KCd
    Jan 29, 2018 at 2:21
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    $\begingroup$ Oh, I see: asq = $a^2$, usq = $u^2$, and so on. I will fix this so it reads more naturally. $\endgroup$
    – KCd
    Jan 29, 2018 at 2:25
  • $\begingroup$ Thanks. Edit in "in other words" for IOW, also. (Would do myself but don't want an edit clash.) $\endgroup$
    – guest
    Jan 29, 2018 at 2:28

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