This is just a straight post, framed around answering the question:
How does one introduce-explain integration by substitution to high school students?
That is the question, but I’ll declare from the outset that I cannot answer it. What I will do is explain as clearly as I can why integration by substitution works in the form(s) in which we use it. It is then up to the teacher to decide how much of this “why” message, if any, is required or helpful for their students. (It is not at all clear to me that delving into the proper “why” of substitution will have much meaning or benefit for more than a few school students.)
The post was motivated by a related request on a recent WitCH. Also, having pondered and hunted through the blog, I notice that frequent commenter SRK made a similar request long ago, and there was a related WitCH. The extensive discussion on those posts may be of interest.
THE BASIC MEANINGS
Just so were all on the same page, the only thing we’re considering on this post are antiderivatives: there is no calculation of areas, no fundamental theorem of calculus. I shall use the term “integration” and integral notation because it is common to do so, but the word and notation properly refer to the summing up of bits, which is not what we’re doing here.
So, the function 
is an antiderivative of 
if 
. We then use the integral sign to represent the general antiderivative:
![\[\boldsymbol{\int f \ = \ \int f(x)\, {\rm d}x \ = \ F + c\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-b29c1f604ea0995f606802b2894704de_l3.svg "Rendered by QuickLaTeX.com")
(Just as a function may be referred to as f(x) or simply as f, the dx notation in integrals is optional, and I’ll use it or not as seems to be clearer.)
An alternative name for this general antiderivative is indefinite integral. Then, the definite integral indicates for us the evaluation of the antiderivative at the “endpoints”:
![\[\boldsymbol{\int\limits_a^b f = F(b) - F(a)\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-3deb41431bdc5dcd689e3388d2a94013_l3.svg "Rendered by QuickLaTeX.com")
Again, there is no “integration” here, no computation of areas. It is almost solely definition and notation. The only substantive point is to recognise that any two antiderivatives of differ by a constant, which is intuitive but takes a proof. Then, this +c, whatever it is, cancels out in the evaluation of the definite integral, implying it doesn’t matter which antiderivative we happened to choose.
STATEMENTS OF INTEGRATION BY SUBSTITUTION
Integration by substitution in indefinite form is standardly presented as,
![\[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x}\, {\rm d}x \ = \int f(u)\, {\rm d}u\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-990ea9c5560ee774f6ec2849c85ad79c_l3.svg "Rendered by QuickLaTeX.com")
Here, on the left hand side, u is some (differentiable) function of x. On the right side, there is a double think: we antidifferentiate thinking of u as any old variable, and then, when done, we think of u as the given function of x.
If we write u = g(x) explicitly as a function of x then substitution takes the form,
![\[\boldsymbol{\int f(g(x))g'(x) \, {\rm d}x \ = \int f(u)\, {\rm d}u\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-6419fb5390f634796c39509c9b2f81e4_l3.svg "Rendered by QuickLaTeX.com")
And, the definite integral version takes the form,
![\[\boldsymbol{\int\limits_a^b f(g(x))g'(x) \, {\rm d}x \ = \int\limits_{g(a)}^{g(b)} f(u)\, {\rm d}u\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-a3b74da7fc4888b10c0d3e5ff07c5b81_l3.svg "Rendered by QuickLaTeX.com")
Note that the definite integral form requires no double-think: the u on the right hand side is simply a who-cares variable of integration. We can also do without x and u entirely, writing the definite integral equation more simply, more purely and less helpfully as
![\[\boldsymbol{\int\limits_a^b (f \circ g) g' \ = \ \int\limits_{g(a)}^{g(b)} f\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-d9db76847f9380e71620057170b90a8b_l3.svg "Rendered by QuickLaTeX.com")
Finally, a quick word on the intermediary, dodgy line:
![\[\boldsymbol{{\rm d}u = \frac{{\rm d}u}{{\rm d}x} {\rm d}x = g'(x) {\rm d}x\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-c76999962d5c9dba395d94a50071daaa_l3.svg "Rendered by QuickLaTeX.com")
Whether or not one permits the dodgy line is really just a detail, since it is immediately followed by a non-dodgy line. It is, however, better to permit the dodgy line, because: (a) it works; (b) it helps; (c) it really annoys people who object to it.
JUSTIFICATION OF INTEGRATION BY SUBSTITUTION
At its heart, of course, integration by substitution is simply the chain rule in reverse. The chain rule for the composition 
is,
![\[\boldsymbol{(F \circ g)'(x) = F'(g(x)) g'(x)\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-2a656c6fd0c21f884a417806067f5de3_l3.svg "Rendered by QuickLaTeX.com")
The chain rule can then be written in antidifferentiation form as,
![\[\boldsymbol{\int F'(g(x)) g'(x)\, {\rm d}x \ = \ F(g(x)) + c\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-b1284b1ac11913f58b8eea3c33778f19_l3.svg "Rendered by QuickLaTeX.com")
Or, with u = g(x), we can write the anti-chain rule as,
![\[\boldsymbol{\int F'(u) \frac{{\rm d}u}{{\rm d}x}\, {\rm d}x \ = \ F(u) + c\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-f2139dca318fd574d3906bf30319e6a6_l3.svg "Rendered by QuickLaTeX.com")
But now, given an integral of the form 
, it is easy to apply the anti-chain rule. All we need is to give a name to the antiderivative of f.
So, let’s write F for the (an) antiderivative of f: that is, F’ = f. Then, by the anti-chain rule,
![\[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \ \int F'(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \ F(u) + c \,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-66c489f805928a8eec9973d13a231e41_l3.svg "Rendered by QuickLaTeX.com")
But also, just thinking of F as a straight antiderivative of f, we have,
![\[\boldsymbol{\int f(u) \, {\rm d}u \ = \ F(u) + c \,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-38e40bec0fb3519d6db745f9927aea46_l3.svg "Rendered by QuickLaTeX.com")
Combining the two lines, and keeping in mind we think of u = g(x) after antidifferentiating, we have integration by substitution:
![\[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \int f(u) \, {\rm d}u\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-6f11ed2d71b90c1b700d19af93175f23_l3.svg "Rendered by QuickLaTeX.com")
The other forms of the formula can be thought of and derived similarly. For example, again setting F’ = f, the definite integral form can be justified as follows:
![\[\boldsymbol{\int\limits_a^b (f \circ g) g' \ = \ \int\limits_a^b (F'\circ g) g' \ = \ \int\limits_a^b (F\circ g)' \ = \ F(g(b)) - F(g(a)) \ = \ \int\limits_{g(a)}^{g(b)} F'\ = \ \int\limits_{g(a)}^{g(b)} f\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-6d58588af66348fb19ec6e4a8f84cd9d_l3.svg "Rendered by QuickLaTeX.com")
CONCLUSION
Will this help? Probably not: the introduction of (and then disappearance of) the antiderivative F is not so easy to understand. So, it is not necessarily wrong to take a “looks kinda right” Leibniz shortcut, or to focus upon a specific chain rule or two. But, ideally, teachers should have some sense of why things are true, even if they then decide to not try to convey this sense to their students. And the sense, as best as I can express it, is the above.
Is it a fair summation (for the majority of Methods students who do not do Specialist and will be taught this only to overcome the stupid formula mentioned recently) to say the following:
1. We can use the chain rule in reverse (as it were) to integrate this because we would use the chain rule to differentiate it?
2. This next line is cheating, but it helps clarify what is happening, so do it?
i think it’s fair enough for both Methods and Specialist students, but others may disagree. But of course the practical aspect is that you can use substitution even if you don’t know exactly how or whether it’s gonna help: you don’t need to see the antiderivative F ahead of time.
RF,
I agree with your comments.
Integration is often hit and miss ….but differentiation is applying a set of rules
https://xkcd.com/2117/
Steve R
Leibniz’s notation for the derivative looks like a fraction – but isn’t. However, it is suggestive.
I wonder if this was Leibniz’s intention.
I don’t know the history, but it’s hard to imagine otherwise.
At the risk of going off on a tangent (!), one may want to note that the integral notation is also by Leibniz; and I think that moreover, he realized that his notation for differentiation and integration was highly suggestive as a whole, given the interplay between the two. The second volume of “Einführung in die Höhere Mathematik” [Introduction to Higher Mathematics, reviewed in Mathematical Reviews; in German] by Karl Strubecker has, as far as I remember, a historical quote by Leibniz on the reason for why to add the differential
(or 
or whatever the name of the variable) in the notation for the integral. Unfortunately I cannot find the Leibniz quote online.
Thanks, Christian. This is why some more pure texts, including school texts, will write A(f) or similar for the antiderivative, rather than ∫f. I thought of that, but I figured pretty much no one is gonna read this post as is: using A(f) would probably have resulted in negative readers.
Q: How does one introduce “thing”.
A: By “imitation and practice” (hat tip Aristotle). And start with an easy example. Get the drilling going right after. And gradually go to harder ones.
Along the way make sure to get them to change the dx also. But more in the spirit of gotta be diligent and change the x any damn place it’s hiding…or will get wrong answers. Not in the baby real analysis rigor mode.
Routinely, I see people with advanced math, think that the way to introduce topics is by rigorous deriviation. This is wrong. That can come later, or during. But not first. First is familiarization. We are meat monkeys, not rule based computers.
Is anyone here suggesting otherwise?
You ask how to introduce topic (as a rhetorical question, maybe?) and I answer.
I do appreciate that you expressed some reservations about introduction by derivation of a general case. But then you go and write one down. I am expressing the (important) opinion that pedagogy is the key to how to introduce. Not mathematical justification.
Here’s what I want:
“OK, class…what if you had to integrate this?
[Show example of hard looking integral of a function they haven’t learned how to. ]
“Well…we don’t know how to integrate that!”
“But we do know how to integrate a polynomial. We love those. They are easy-schmeasy!”
“Wouldn’t it just be dreamy, if we could transform this hard integral, we don’t know how to solve into a simple one, like a polynomial, or something else that we can at least handle?”
“Well…it turns out…that sometimes you can. It’s a really cool trick called integration by substitution. That’s right…INTEGRATION BY SUBSTITUTION. That’s what we call this sneaky trick!”
“Let me show you how that works, here.”
[Solve the problem on the board.]
[Have the kids work an immediate similar problem, together, step by step, showing the aspects on their chinned whiteboards…or paper and pencil if you are not that Gregish.]
—
That’s good enough. Then have them do more and more of progressively harder nature.
At some point, but AFTER several “layup” baskets, you can opine a little on how picking a substitution (like integration in general) can be a little bit of a guessing game…and that the key is doing lots of problems, so you can get an intuition about what sorts of substitutions to try.
—-
Your general case writing stuff (in post) may be useful to build their math muscles. But only with the stronger tracked classes (Specialist versus Methods or whatever the names are). And only AFTER, they have familiarized with the technique itself. I.e. NOT as an “introduction”.
P.s. And “it still moves” and “hand me the hemlock”. 😉
“What I will do is explain as clearly as I can why integration by substitution works in the form(s) in which we use it. It is then up to the teacher to decide how much of this “why” message, if any, is required or helpful for their students. (It is not at all clear to me that delving into the proper “why” of substitution will have much meaning or benefit for more than a few school students.)”
“I do appreciate that you expressed some reservations about introduction by derivation of a general case. But then you go and write one down. I am expressing the (important) opinion that pedagogy is the key to how to introduce. Not mathematical justification.”
😛
The sole purpose of this post was to indicate as clearly as I could why substitution works. Your sole purpose here appears to be to ignore the sole purpose of the post.
I’m interested in the first two-clause sentence of the post.
And I disagree with your following sentence, “That is the question, but I’ll declare from the outset that I cannot answer it.” I bet you could opine thoughtfully on how to introduce the topic, based on your actual teaching experience. [You’re just not doing that, here.]
Also, it kind of contradicts your first clause” ‘this is a straight post to answer the question”…to shift into not answering the question! It’s a straight post on why substitution integration works. NOT a straight post on how to best introduce the topic.
And…I won’t be the only rock-kicker, confused by the segue, Marty. I like you. And like to provoke you. But honest, that was a confusing intro. 🙁
I don’t mind you provoking me. I would prefer it if you read me carefully before doing so.
As well as misinterpreting me, you’re misquoting me. What I wrote was, “This is just a straight post, framed around answering the question …”
That “framed around” was a conscious choice. Not a great choice of wording, I’ll admit, but a choice to indicate difficulty with the nature of the post.
Why? Because, yes, teachers want an answer to that question but, no, I cannot answer the question. The answer depends upon the teacher, the students, the assessment, and a bunch of other contingent facts. And this isn’t just hypothetical: any proper explanation of substitution is far enough removed from the magical mechanics that it in practice the appropriate approach will vary hugely.
I then state, “What I will do is explain as clearly as I can why integration by substitution works in the form(s) in which we use it.”
That seems clear enough to me, whatever you thought about the lead-up.
You wanted to have a go at (some) “people with advanced math”. Fine, and I’ll pass you the rocks, if and when it is called for, which is plenty often. It wasn’t called for here. You just misread me and charged into an imaginary battle.
1. The issues with the introduction would remain, even if I had cut and paste quoted “framed around”. The post is not framed around answering that question. It’s framed around NOT answering that question and deciding to answer a different one! And it’s not “straight” to shift the topic like that.
2. More importantly, I disagree with the assertion that one cannot say important things about how to introduce the topic. For example, rigor versus example. Sure, there are a multitude of students and they may benefit from different approaches. But teachers still have to make practical decisions on how to introduce the topic. Just because it’s not a Euclidean proof, doesn’t mean there aren’t better approaches than others. To whit, I’d argue that eschewing (forever, or at least during the intro) your derivation is a good approach. And leading with it is a bad approach. And I’m NOT saying that you said to lead with it. I’m saying that it IS possible to discuss meaningfully, the initial question, which you used as a hook and then diverted from.
For an anonymous person commenting on a free blog, you’re mighty self righteous.
Sure, yeah…I’m a piece of work. But, of course, the points stand. 😐
It’s OK. I’ll buy you a beer even after you ban me–you’ve got balls. Of course, this implies going south of Sydney…and all the NSW/QLDers told me that was completely pointless. 😉
After reading the comments, I understand why they advise me to never read the comments.
Hard to disagree. Although I think on some posts the comments are a little more commenty.