# which exact integration techniques belong in a first year calculus/analysis course?

At our university we are now discussing changes to the course contents and there is some heated discussion regarding integration in the first year calculus courses. Currently, the techniques of exact integration include integration by parts, substitution, and lots of quite complicated formulas for the integrals of various trigonometric functions, and quite some emphasis on tricks for trigonometric substitutions, and integration by partial fractions.

I'm of the opinion that more emphasis should be placed on properly understanding integrals and only drill the most elementary techniques: by parts, substitution, partial fractions. I personally see little use for being able to compute complicated integrals by hand when a computer will do it in a split second. I see integration techniques disappearing from the standard tool-box of the mathematician, replaced by software. I thus see integration techniques as a niche and think that students should not be wasting much time on perfecting their integration skills. They just need to be able to solve simple integrals by hand.

I'm interested in hearing other opinions and, particularly from students, which integration techniques that you learned in your first year do you actually find useful in your later studies (whatever your discipline is).

Thanks!

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While the issue addressed is of the utmost interest, I'm not convinced the question as posed can receive good answers, as opposed to opinions and anecdotes. I'm eager to be proven wrong, though. For my own two cents I think there's plenty of conceptual content to, for instance, trigonometric substitution, but that memorizing formulae for it is an utter waste of time. – Kevin Carlson Jan 31 '14 at 23:10
good point @KevinCarlson, so I added the big-list tag. – Ittay Weiss Jan 31 '14 at 23:41
The axioms of $\sf ZFC$, proving the relative consistency of $\sf ZFC+GCH$ and $\sf ZFC+\lnot CH$. If there is time left, then large cardinals, and the constructions of "Violating $\sf CH$ with a single real". :-) – Asaf Karagila Jan 31 '14 at 23:43
I think there is some value to certain trigonometric forms, but formula memorization is useless, especially when so many students fail to understand, for instance, periodicity or multivaluedness. – Emily Feb 1 '14 at 0:21
Lecture halls with hundreds of students, and help seessions manned by graduate students are outdated. Put the best lectures on youtube and let people ask questions on stackexchange. University methods for teaching elementary classes are outdated. – TrialAndError Feb 1 '14 at 2:10

I am a 4th year math major with a computer science minor at a University, and do quite a bit of research in digital signal processing as a hobby (so that might bias my answer somewhat). I can say that integration by parts has been BY FAR the most useful technique I've learned. I can also say that after Calculus II I never used any of the trigonometric integration tricks. And I think that substitution is also very important. Even though I don't use it very often to solve integrals it has made me immeasurably better at seeing solutions to equations in general, and in constructing proofs. Anyway, just my 2 cents

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For signal processing, partial fraction decomposition is definitely the go-to method since most involved systems are composed of functions that can be designed or described that way. I learned that in an engineering class though, after having my brains scrambled by a high-variance distribution of math professors ;) – Trevor Alexander Feb 1 '14 at 4:16
I feel your pain. +1 for partial fraction decomposition. Also, just a straight-up 2nd or 3rd order Taylor approximation will often be more than sufficient for a lot of systems. I didn't include that because I feel like the question was geared more toward "pure" math rather than applied math or engineering – Chad Russell Feb 1 '14 at 4:25
Thanks for the answer @ChadRussell. My question is not meant to be exclusive for pure maths. On the contrary in fact. Your input is helpful! – Ittay Weiss Feb 1 '14 at 10:16

Though I can't speak for a calc student ( being self taught in derivatives and integration), personally I think that first year integration should include power rule, integration by parts, and integration by substitution. Second year continuing with the more complex techniques. Being neither a formal student nor a teacher, however, my opinion a little uninformed.

As for turning over manual calculations to a computer/calculator I can provide the perspective of a Statistics student whose prof completely abused the programming capability of the TI-84. I walked out of that class with a B average and retained nothing... Why would I? The calculator did all the work. I didn't even have to write my own programs. I just punched the numbers into the calculator and presto change-o I got a good grade.

Why teach manual integration when a computer will do it for you? For one, there are integrals that can't be written in a nice form and give a computer fits. But more important than that is an intimate understanding the process, something you can not attain if you forsake manual learning for the algorithm.

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Thanks Chris. I just wanted to clarify that I did not suggest not teaching any integration techniques, but to emphasize more of what the integral is at the expense of focusing less on integration techniques. By that I hope to enhance the intimate understanding of the process. – Ittay Weiss Feb 1 '14 at 10:14
The funny thing is I never liked math as a student. I had a nasty habit of changing signs during a test which pretty much doomed me to a C or D. It was only after in my own adventures that I found a use for and then an interest in the subject. But what I've found is that lacking some of the background ( Trig proofs and identities being my major obstacle) At times I feel I have the fluency of a stuttering 2 yo esl student. It hasn't curbed my interest esp. since I love reading physics books. But it does make it a devil of a time to understand them – Chris Feb 1 '14 at 13:42
When I am not fluent in the "language" used. I can understand the text, and the conclusions but my intimacy with the mathematical reasons is lacking because I am not used to following the process. So to me it is not enough to relegate myself to a trained monkey pushing buttons I need to understand the manual process to understand the books/papers I like to read. So do your students. I guess its like taking a gun into bear country. Would you rather have it and not need it or need it and not have it? – Chris Feb 1 '14 at 22:07

I think any meaningful answer would necessarily depend on the intended audience for such a course. Many universities offer different levels of calculus courses, ranging, for example, from "calculus for non-science majors" to "calculus for math majors." Who is such a course intended to serve? Business/Economics majors might want to know basic concepts like derivatives and integrals for their future coursework, but a calculus course for an electrical engineering, physics, or math major is going to want to treat the material at a much deeper level.

Then, of course, we should also consider the caliber of the institution: A state college would not be expected to have the same academic standards as, say, Harvard, MIT, Caltech, Stanford, or Princeton mathematics.

At this point, I should mention that teaching techniques of integration for higher-level students is not simply about teaching practical skills. Everyone who has had at least an undergraduate mathematics education that encompasses analysis, algebra, and geometry (i.e., a bachelor's degree in math) should by now realize that the value of being taught such concepts is not in being able to say at a later date, "oh, I'm in such-and-such a career and I use group theory at my job every week." It is about learning how to think abstractly and logically; to develop broad computational proficiency that allows one to be more efficient in problem solving.

So, without considering those factors, I'd say it's difficult to pin down what would be "appropriate" for a first year calculus course. If I could design my own course without any external considerations, I'd cover as many techniques of indefinite integration as I could think of, including obscure ones, just because they I personally think they are interesting and "fun." But again, teachers, math departments, and universities are not offering these courses for their own entertainment. They are (or at least, should be) offering what they believe will be the most useful skills and knowledge for their students to be successful.

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I completely agree that the value is in learning how to think abstractly and logically; to develop broad computational proficiency that allows one to be more efficient in problem solving. And here is precisely my question: how do the various tricks of integration techniques promote that? If all that is taught is the algorithmic technique (i.e., see that kind of integrand, try that substitution etc.), what does that teach the student? – Ittay Weiss Feb 1 '14 at 10:21
Integration techniques is one of those instances in mathematical computation where students are encouraged to try different things. The student eventually develops a sense of what types of substitutions are most likely to be successful. This later extends to problems that involve integration in practice--e.g., the solution of differential equations; computation of posterior distributions in statistics; quantum mechanics; Fourier analysis. Computers are great, but it is only through direct experience that one develops the intuition to use them effectively. – heropup Feb 1 '14 at 10:31
Then, of course, we should also consider the caliber of the institution: A state college would not be expected to have the same academic standards as, say, Harvard, MIT, Caltech, Stanford, or Princeton mathematics. why shouldn't they? – Chris Feb 1 '14 at 13:51
@Chris Your question implies an institution-centric viewpoint. The fact of the matter is that there must exist institutions with lower admission standards, that aim to serve students that, for example, may not have had calculus in high school. These students aren't going to have the familiarity with mathematical thinking that an undergraduate math major at Harvard is going to have. Of course standards will not be the same, in as much as it is unreasonable to require a business major to know algebraic topology. – heropup Feb 1 '14 at 17:37
@heropup I'll give you that a music major won't need to be taking advanced nth level integral equations, but don't students have the right to demand the HIGHEST quality education for the level of mathematics the aspire to? They are paying for it after all. That is what I am getting at. – Chris Feb 1 '14 at 22:05