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This is the question that was not asked here. Also related is this question, but both presuppose that it will be taught and ask about how best to do it. My question here is, suppose we are designing a calculus curriculum; should we include integration by trigonometric substitution?

Some arguments in favor:

  • It teaches abstract pattern-recognition skills
  • With multiple integration techniques in a "toolbox" the students can learn the experience of being given a problem and having to choose the right tool
  • It's fun to see how an integral that doesn't seem to involve trig at all can be integrated using it
  • It's necessary if you want to actually evaluate by hand any but the most trivial integrals in multivariable calculus (possible rebuttal: why would you want to do that?)

Some arguments against:

  • No one is ever going to use it; in practice nowadays we just reach for a CAS
  • It's hard and difficult to motivate, and only math majors will enjoy or appreciate it; to everyone else it's just one more useless thing to memorize, lowering their opinion of math still further
  • It takes time away from other topics that are more likely to be useful.
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    $\begingroup$ To counter the "just reach for a CAS" argument: Often, they will give you an answer which involves erf or hypergeometric series for things which have nicer forms than that. $\endgroup$ – Adam Oct 3 '15 at 0:13
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    $\begingroup$ @Adam, will you give an example of a CAS answering using erf when there are nicer forms available? $\endgroup$ – user173 Oct 3 '15 at 3:41
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    $\begingroup$ The basic issue is that the audience for a calc course is too broad. It doesn't make sense to offer calculus as a one-size-fits-all class. This more true than ever because so many biology departments, in an effort to weed out prospective majors, require their students to take calculus and calculus-based physics. It's totally absurd to be force-feeding trig substitutions to people who want to be dentists and pharmacists. $\endgroup$ – Ben Crowell Oct 3 '15 at 18:21
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    $\begingroup$ @BenCrowell I agree, but at my university I don't think there are enough calc 2 students to enable us to split it into separate trarcks. $\endgroup$ – Mike Shulman Oct 4 '15 at 11:51
  • $\begingroup$ (1) You can make the same, "just use a CAS" for even more basic techniques. (2) The everyday student doesn't "just use a CAS" as much as people think here. In fact many users of CASes are rather sophisticated themselves. (3) Many physics problems use integrals that you can just look up, but if you know trig substitution, you know where the all came from (even if you refer to a table). $\endgroup$ – guest Oct 23 '18 at 0:53
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In reality, I think this is not the most important topic, and if I was designing a curriculum from scratch, I would probably omit it.

We rarely have that luxury however. In my state, for example, all state colleges must meet certain common standards to be able to transfer credit: one of these standards is teaching trig sub.

If I had to include it, I think that teaching trig sub could be valuable in the context of a larger story.

You could tell them how not every function is integrable in terms of elementary functions. Then a natural question is "Can we characterize all those functions which are?". This is too hard for this beginning course. However, we can at least find some large families of functions which we can integrate. Certainly all polynomials. With some work, all polynomials in $\sin$ and $\cos$.

Another big class to try and tackle is rational functions. At the start of the day, there is no reason to think these should all be integrable in elementary functions. But the fundamental theorem of algebra tells you that you can factor denominator as a product of linear terms and irreducible quadratic terms. Partial fraction decomposition further simplifies. Finally, we need trig sub to knock out the terms with quadratics.

So trig sub becomes valuable as part of a story about how to integrate any rational function.

Once you know how to integrate any rational function, it turns out you can also integrate any rational function of $\sin$ and $\cos$ using the tangent half-angle substitution, which has connections to some basic algebraic geometry (Rational parameterization of the unit circle).

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  • $\begingroup$ Here we have the best reason, as a means to introduce them to algebraic geometry. But, how will we show that finding all the rational points on a circle is applicable to the real world? $\endgroup$ – James S. Cook Oct 8 '15 at 3:20
  • $\begingroup$ Finding all the rational points on a circle is a lot easier than this, I think, or at least I do it without integration when teaching number theory ;) but I do like the idea of introducing the ring of periods to freshmen. $\endgroup$ – kcrisman Oct 13 '15 at 13:06
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    $\begingroup$ @kcrisman The rational parameterization given by mapping $y$ to the point on the circle connecting $(-1,0)$ to $(0,y)$ is the "tangent half angle substitution". The connections to integration are not needed in a number theory course, but they are there! In fact, classically I believe it was the inability to find such a method for integrals like $\int \frac{1}{\sqrt{x(x-1)(x-\lambda)}} dx$, which lead Riemann to discover his surfaces, and to see that the obstruction was topological (One cannot rationally parameterize a torus)! $\endgroup$ – Steven Gubkin Oct 16 '15 at 4:29
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My first feeling on this is that, although in any given calculus course there might be a reason not to introduce trig substitution, in practice it is still a valuable technique that takes a whole huge class of integrals and "make sense" of them. For instance, to tell whether you have just fed your CAS junk - misplace an $x^2$ by and $x^3$ and it's all over! So rather than being reliant on a black box for all these things with square roots of $x^2$ you at least know why you should expect logs or other such stuff as an answer. I find it hard to believe it's hard to motivate - surely there are many real-life physics integrals which need precisely this technique? Naturally it might not be quite as fun for econ or bio majors.

Myself what I like to do is give some 'greatest hits' of it. But perhaps it's not so useful to spend two very long days of class doing it any more, that is true.

I also have a suspicion (untested) that even the most powerful CASes might have some situations which their pattern-matching does not recognize as a trig substitution where a human might. It would be interesting to see some such examples.

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    $\begingroup$ Knowing what sort of thing to expect from a CAS so that you can recognize errors is another good argument in favor. $\endgroup$ – Mike Shulman Oct 4 '15 at 11:49
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    $\begingroup$ On the untested suspicion, see some anecdotes at reddit.com/r/math/comments/2taw1s/…. My contrary suspicion is that a CAS's failure to recognize trig substation is much less likely than a human's failure to recognize or failure to apply it correctly. $\endgroup$ – user173 Oct 4 '15 at 15:52
  • $\begingroup$ Haha! True. Note I said "some situations", not "most" or even "more than a small number". They say that hundreds of errors were discovered in integral tables when the first CASes were developed - I assume this is not just a legend. $\endgroup$ – kcrisman Oct 5 '15 at 12:17
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Yes. We should teach trigonometric substitution. But, I take it a step further, I think we should also teach hyperbolic substitution. With this additional technique the idea of the substitution is much clearer. Also, teach it with confidence. Teach it as if they can all understand it... because they can.

Why do students have trouble with this topic? I submit, the reason is often ignorance of trigonometry. Especially an inability to manipulate basic algebraic identities. Should we face this inequity of their past education in second semester calculus? I think yes. I could throw up my hands in despair and just say they're too far gone, but, I refuse. It's not too late to learn elementary algebra and trigonometry. That is all trig-substitution entails.

Almost any topic we cover in calculus, or algebra for that matter, can be done with a CAS. However, to think that the point of calculus is to solve calculus problems misses the mark. The real reason is that these students (I am not talking about the life-science calculus crowd here, rather those students destined for science-heavy technical majors or math) need to refine their algebra skills. When they rely on a CAS to work around it, they make the course a waste of time. Much like a calculus III student I talked with today who is using a CAS to do his homework, fine, but, I wonder how will the test go?

Basic knowledge of graphs of sine, cosine, tangent etc as well as their values is in short supply with many of this generation. I believe this is in no small way tied to overuse of calculators in the precalculus mathematics. On the other hand, it also tied to the fact that their education has been eviscerated by the removal of more and more topics which we deemed too difficult by "educators". Surely, by now it has been clearly demonstrated that the removal of hard topics from elementary school does not bring us a society of like-educated people. In fact, with lower standards we all get left behind because none of us really know the math anymore. I've seen the texts my colleague used as a child. It's sobering to see how far we've allowed the standards to drop.

Hold the line. Teach trigonometric substitution. If nothing else, as an opportunity to teach some much needed trigonometry in calculus.

I should qualify, I do not view university education as a job-training program. I want my students to think. If they want to solve problems without thinking later, that's their business. The reason we solve problems in calculus is not for their applicability. Rather, we solve problems in the interest that the student learn to think. If the problem happens to be really really real-world applicable then great, but, that is not the primary purpose.

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    $\begingroup$ The very real danger is that there are so many essentially mechanical techniques which must be mastered, and so few core concepts, that the tests become tests of mechanics, and the "thinking" in the subject is lost. I am all for dropping trig sub, partial fractions, even integration by parts if it allows me enough time to drive home the meaning of the fundamental theorem of calculus (for instance). $\endgroup$ – Steven Gubkin Oct 8 '15 at 3:26
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    $\begingroup$ @StevenGubkin Sure, if I have 3 hours to talk to folks about integral calculus, then I'd just focus on polynomials, u-subst. and the FTC. However, I have a semester. There is time for both integration theory and techniques(IBP, par-frac, trig-subst,...). $\endgroup$ – James S. Cook Oct 8 '15 at 13:43
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    $\begingroup$ I think it takes much longer than three hours to get across the basic points of integration theory. I also think there is a whole lot of creative stuff we could be getting from students, opposed to learning these integration techniques, which have very little creative aspect. $\endgroup$ – Steven Gubkin Oct 9 '15 at 18:45
  • $\begingroup$ @StevenGubkin It becomes creative when you ask them questions you have not already done. I completely advocate asking open ended questions, or questions with infinitely many outcomes, etc. I suppose calculus is not taught this way as much as say physics. I know when I teach physics there is no possible way I can cover how to do all problems. What I can do is to present the basic physical laws and illustrate with a few examples. The same goes with these substitution problems, show them a few and hold them accountable for the concept. Ideally, they hone trig/algebra skill in the process. $\endgroup$ – James S. Cook Oct 9 '15 at 18:59
  • $\begingroup$ But, I should mention, if I had the choice between teaching trig-subst. and giving a talk about the active creative aspect of modern mathematics. If that was the choice, I'd totally go for advocating mathematics as alive. For me, this is not a choice I have to make since I weave the unfinished nature of math into the discussion whenever I can. For trig. susbt. saying a word or two about the connection with algebraic geometry might be very interesting for the best students. $\endgroup$ – James S. Cook Oct 9 '15 at 19:02
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I think teaching trigonometric substitutions is a good idea - that is, I presume you mean working out integrals such as

$\displaystyle\int\frac{1}{x^2+a^2}\mathrm{d}x$ and $\displaystyle\int\frac{1}{\sqrt{a^2-x^2}}\mathrm{d}x$?

I think learning to work out integrals such as these is useful from a mathematical point of view, because it exposes students to the idea that an integral can be evaluated by using a substitution of the form $x=x(t)$ rather than using $u=u(x)$. Then students can calculate a greater class of integrals, and they need to more carefully consider the limits and what $\mathrm{d}x$ is in terms of $\mathrm{d}t$.

These integrals also emphasise the importance of calculating the limits, which involves needing to take inverses of the function used for the substitution, rather than just the function itself.

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    $\begingroup$ the connection with more experience with changing bounds is a good point. In my experience, learning to change bounds is a difficulty topic for about half the students. $\endgroup$ – James S. Cook Oct 8 '15 at 3:23
  • $\begingroup$ Indeed. It reinforces knowledge of inverse functions, so if someone wants to be able to effectively evaluate an integral with a hyperbolic substitution, then they need to know what the inverse hyperbolic functions are. $\endgroup$ – omegaSQU4RED Oct 8 '15 at 15:30
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Just to show what it could be like, Math 116 at Michigan is a Calc II course which does not cover trig integrals, and which I consider one of many possible good courses. You can read the syllabus here which, combined with the table of contents for the textbook, should give you an idea of what is covered. Chapters 7.3 and 7.4 are what I think of as the standard "techniques of integration" material, and they are not included.

Here is the most recent final exam, to give an idea of what students learn to do by the end of the course.

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  • $\begingroup$ I like Question 8a in the exam for techniques of numerical integration, and I like the clarity of the questions and the answer-forms. $\endgroup$ – user173 Oct 10 '15 at 0:41
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I taught a calc II course at my university using Mathematica for evaluating integrals. It allowed us time to consider a more vast array of applications of integrals and the students learned more about how to model a problem using an integral, since we saved so much time that would have been wasted evaluating integrals. Besides mentioning a few numerical integration techniques and their error estimates, I didn't teach how numerical integration really works. Student feedback was overwhelmingly positive and I was very happy with the final exam performance. I will definitely do it again. In short, to answer the OP, I say, don't teach trig substitution.

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Not necessarily an answer, but some points:

  • If further courses require students to be able to do trigonometric substitution, then it does need to be taught; the calculus curriculum may or may not be the place to do it. (but one should probably ensure that it will happen elsewhere before one excises it from the calculus courses!)

  • If one expects students to rely on CAS to solve problems that require trigonometric substitution, then there must be a place in the curriculum where students learn how to use CAS effectively.

  • In order to get the most use out of CAS, it helps to have an idea of what sorts of things they can solve; both so that they know what to aim for when analyzing a problem, and where they can stop doing analysis and just solve.

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No, we should not teach trigonometric substitution. Your arguments against are overwhelming.

You can teach a toolbox of techniques for numerical integration instead. E.g.: Find three ways to answer to an accuracy of .01, what is the area of the region with $\sin(x)>y^2$ and $|\,x\,|<\pi$?

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    $\begingroup$ I didn't downvote, but it's unclear to me that a toolbox of techniques for numerical integration is any more useful. Surely a CAS is at least as good at numerical integration as it is at symbolic integration. $\endgroup$ – Mike Shulman Oct 4 '15 at 11:49
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    $\begingroup$ @MikeShulman, of course CAS are good at numerical integration. But symbolic integration is not a particularly transferable skill. By contrast, Monte Carlo numerical integration is a good introduction to Monte Carlo techniques generally. Numerical integration of unbounded functions or on infinite regions is a good introduction to error estimates and issues in analysis. Choosing between techniques in simple cases is a good introduction to choosing when you need to limit calculation time or when your inputs are approximate. Numerical integration can get students thinking in useful ways. $\endgroup$ – user173 Oct 4 '15 at 16:10
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    $\begingroup$ On the contrary, pattern matching and creative reorganization is also "transferable" in some general sense as well. And who is teaching Monte Carlo numerical integration in a first course where trig substitution is an expected technique? (I also wouldn't downvote this answer, though, it's not unhelpful, I just disagree with it.) $\endgroup$ – kcrisman Oct 5 '15 at 12:19
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    $\begingroup$ I downvoted. I disagree the agaisnt points are overwhelming. I disagree that nobody uses it nowadays, nor that it is hard to motivate. Most simple electrostatics integration problems have to be solved by trigonometric substitutions. I am from Brazil, and most people don't reach for a CAS, and those who do it do because they are incompetent at using math tools. Takes time from what that could be more useful? What else is in the calculus program that could be more useful, taking in account the arguments above? $\endgroup$ – Mark Fantini Oct 5 '15 at 14:09
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    $\begingroup$ I would like to see a discussion of why numerical integration is a good substitute, not a vague sentence about teaching a toolbox of techniques for numerical integration and a question whose relevance over trigonometric substitution is not apparent nor overwhelming. I don't mean to be rude, just straight. $\endgroup$ – Mark Fantini Oct 5 '15 at 14:14

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