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This is a question about notation. Suppose I'm giving a worked solution for a definite integral which involves a substitution, and I want to show an intermediate step which includes both the original integrand and the substituted variable.

For example, with the substitution $u = \sin x$:

$\int_0^{\pi/2} \sin^2(x) \cos(x) dx$

$= \int_{0}^{???} u^2 \frac{du}{dx} dx $

$= \int_0^1 u^2 du$

In the middle line where both $u$ and $x$ appear, should the upper limit be given as $\pi/2$ (i.e. defined in terms of $x$) or 1 (i.e. in terms of $u$)? (The lower limit will be 0 either way.)

My interpretation is that the final $dx$ in this expression means it's still an integral in terms of $x$ and so the limits should still be in terms of $x$. But I'm editing some material that does it the other way, and I want to be sure I'm not "correcting" something that's already correct.

If my interpretation is correct, a citeable reference would be useful.

GB supports the mod strike
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    Since you do not consider $u$ as a function but as a new integration variable, the middle line is a heresy which cannot be healed. Just wipe it off. – Anne Bauval Apr 07 '23 at 08:40
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    In class and on handouts and such, I used to write things like $;\int_{x=0}^{x=\pi/2} u^2 \frac{du}{dx};$ and $;\int_{u=0}^{u=1} u^2 \frac{du}{dx};$ and $;\int_{x=0}^{x=\pi/2} u^2 du;$ -- whatever seemed most appropriate and clear for the intermediate steps, although I very rarely used either of the first two of these. – Dave L. Renfro Apr 07 '23 at 08:47
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    @AnneBauval Unfortunately not an option - this is worked material where the intermediate steps need to be shown for the benefit of learners, and my scope for making changes is limited. – GB supports the mod strike Apr 07 '23 at 09:01
  • @DaveL.Renfro hmm, that might be a good option here! If we don't get something more authoritative, would you consider posting that as an answer? – GB supports the mod strike Apr 07 '23 at 09:05
  • @Geoffrey Brent: Your comment got me wondering if I've done this in a MSE answer, since I don't recall dealing with definite integrals. Then I thought of this answer, where you can see the use of this method. This notation is certainly not new to me -- I'm pretty sure I picked it up from seeing it in books back when I was an undergraduate (late 1970s). – Dave L. Renfro Apr 07 '23 at 09:33
  • Until you change the variable of integration from $x$ to $u$, the integral - and its limits - are in $x$, and $u$, if present, is only present as a function of $x$. Only when the variable of integration is changed to $u$ should the limits also be changed to $u$ values. In that case, any lingering $x$ around is now to be interpreted as a function of $u$ (and if such an interpretation doesn't make sense, then you've goofed up and should take a more cautious approach, such as Dave L. Renfro's). – Paul Sinclair Apr 08 '23 at 21:27
  • @PaulSinclair I agree with all this, I guess the question is at what point the notation does imply a change to the variable of integration. My interpretation is that as long as the integral expression ends in $dx$, we are still integrating in $x$; the original author's interpretation seems to be that the variable changes as soon as $u$ is introduced. – GB supports the mod strike Apr 09 '23 at 04:24
  • I would default to your interpretation, but I'm also not sure if I've ever seen something like your middle integral, so definitely don't have a source. – Mark S. Apr 10 '23 at 01:53

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