This post is about Dallas, which, it must be noted, is one of the most sterile, soulless cities in existence. As such, it is obligatory to begin with the excellent Dallas, Texas performed by the very excellent Austin Lounge Lizards:
This is our post for discussion of the 2023 NHT Specialist Mathematics exams, which have now been posted, here and here. We haven’t looked, and don’t particularly intend to, unless something is flagged.
This is our post for discussion of the 2023 NHT Mathematical Methods exams, which have now been posted, here and here. We haven’t looked, and don’t particularly intend to, unless something is flagged.
This has a lot in it: it’s more of a coven than a WitCH. We couldn’t see what else to do.
As with our recent PoSWW, this WitCH comes from the Logic and Proof chapter of VicMaths, Nelson’s Specialist Mathematics Year 12 text. This is a new VCE topic, for which the summary from VCAA’s study design (Word, idiots) is,
This summary is also given as the prompt of Nelson’s chapter. Then, the extended excerpt here is Nelson’s introduction to the chapter, together with a few of the associated exercises + answers. (The textbook then continues with its coverage of “conjecture” and so forth.)
Readers who lose sight of land may wish to refer to a solid introduction to argument to get their bearings.
Or a mule, maybe. You can lead a mule to textbooks.
About a month ago, there appeared a Conversationarticle by Rachel Marks, a researcher in primary education at the University of Brighton, in England. Based upon research for which she was the Principal Investigator, Marks’s article was about a UK government program, launched in 2016, where primary schools were offered matching funds to purchase mathematics textbooks. Marks and her colleagues concluded that schools substantially rejected the program: few schools took up the offer of subsidised texts, and fewer stuck with it. Continue reading “You Can Lead a Horse to Textbooks”→
We’ve been meaning to post this for a while, but simply haven’t gotten around to it. Mysterious commenter texas asked a number of “Where do I look?” questions. For one, on who should people read on maths ed, we put up a discussion point, which seemed to garner plenty of interest. So, finally, here is post on the second question (of three). Continue reading “What Are the Good Mathematics Curricula?”→
Last week, Meta-Facebook-Instagram announced a program to combat “misinformation” during the referendum for the Voice to Parliament. This is a bad move, if for no other reason that it will be viewed, probably correctly, as the silencing of voices in order to support the Voice. The ironic anti-message will undoubtedly be clear to Voice sceptics. We had thought to write on this, but figured we’d already written enough on inappropriate Voice spruiking, and on the dangers of half-wit authorities declaring what is or is not misinformation.
M-F-I’s misinformation program is not just a bad move, however, for unintentionally screwing up the Yes guys on the Voice. M-F-I claims that their program is “contributing to democracy”, but there is a solid argument that they’re doing the exact opposite. Which brings us to Matt Taibbi. Again.
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:
(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”:
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,
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,
And, the definite integral version takes the form,
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
Finally, a quick word on the intermediary, dodgy line:
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,
The chain rule can then be written in antidifferentiation form as,
Or, with u = g(x), we can write the anti-chain rule as,
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,
But also, just thinking of F as a straight antiderivative of f, we have,
Combining the two lines, and keeping in mind we think of u = g(x) after antidifferentiating, we have integration by substitution:
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:
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 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")
![\[\boldsymbol{\int\limits_a^b f = F(b) - F(a)\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-3deb41431bdc5dcd689e3388d2a94013_l3.svg "Rendered by QuickLaTeX.com")
![\[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x}\, {\rm d}x \ = \int f(u)\, {\rm d}u\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-990ea9c5560ee774f6ec2849c85ad79c_l3.svg "Rendered by QuickLaTeX.com")
![\[\boldsymbol{\int f(g(x))g'(x) \, {\rm d}x \ = \int f(u)\, {\rm d}u\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-6419fb5390f634796c39509c9b2f81e4_l3.svg "Rendered by QuickLaTeX.com")
![\[\boldsymbol{\int\limits_a^b f(g(x))g'(x) \, {\rm d}x \ = \int\limits_{g(a)}^{g(b)} f(u)\, {\rm d}u\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-a3b74da7fc4888b10c0d3e5ff07c5b81_l3.svg "Rendered by QuickLaTeX.com")
![\[\boldsymbol{\int\limits_a^b (f \circ g) g' \ = \ \int\limits_{g(a)}^{g(b)} f\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-d9db76847f9380e71620057170b90a8b_l3.svg "Rendered by QuickLaTeX.com")
![\[\boldsymbol{{\rm d}u = \frac{{\rm d}u}{{\rm d}x} {\rm d}x = g'(x) {\rm d}x\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-c76999962d5c9dba395d94a50071daaa_l3.svg "Rendered by QuickLaTeX.com")
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")
![\[\boldsymbol{\int F'(g(x)) g'(x)\, {\rm d}x \ = \ F(g(x)) + c\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-b1284b1ac11913f58b8eea3c33778f19_l3.svg "Rendered by QuickLaTeX.com")
![\[\boldsymbol{\int F'(u) \frac{{\rm d}u}{{\rm d}x}\, {\rm d}x \ = \ F(u) + c\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-f2139dca318fd574d3906bf30319e6a6_l3.svg "Rendered by QuickLaTeX.com")
, it is easy to apply the anti-chain rule. All we need is to give a name to the antiderivative of f.![\[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \ \int F'(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \ F(u) + c \,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-66c489f805928a8eec9973d13a231e41_l3.svg "Rendered by QuickLaTeX.com")
![\[\boldsymbol{\int f(u) \, {\rm d}u \ = \ F(u) + c \,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-38e40bec0fb3519d6db745f9927aea46_l3.svg "Rendered by QuickLaTeX.com")
![\[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \int f(u) \, {\rm d}u\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-6f11ed2d71b90c1b700d19af93175f23_l3.svg "Rendered by QuickLaTeX.com")
![\[\boldsymbol{\int\limits_a^b (f \circ g) g' \ = \ \int\limits_a^b (F'\circ g) g' \ = \ \int\limits_a^b (F\circ g)' \ = \ F(g(b)) - F(g(a)) \ = \ \int\limits_{g(a)}^{g(b)} F'\ = \ \int\limits_{g(a)}^{g(b)} f\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-6d58588af66348fb19ec6e4a8f84cd9d_l3.svg "Rendered by QuickLaTeX.com")