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

Question

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!

Answer 1

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

Answer 2

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.

Answer 3

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.