I think if you can devise a methodical approach that benefits students, by all means teach it to them. Understanding your approach would probably teach them plenty.
Okay ... thank you for green-lighting me, Ralph.
Warning: seasoned Calculus teachers are the target audience of this post; new students are not.
Mostly what I had in mind was a methodical approach that would help with those trickier, more tedious, indefinite integrals. Usually I'm able to work along in the standard textbook and explain everything else to them, including most of the concepts like functions and limits, in the standard way that those concepts are taught. But I have found the integration techniques are often poorly explained.
In the book I just picked up from the Salvation Army store, James Stewart's Calculus
(4 ed.), he talks about 4 principal techniques
for finding anti-derivatives:
(1) Immediate recognition of what certain basic functions integrate to.
cos x --> sin x + c
x^2 --> (x^3) / 3 + c
e^x --> e^x + c
(2) Simple substitution.
(3) Integration by parts.
(4) Fancy trigonometric substitution.
Section 5.5 through Chapter 8 of the book covers the topic of "how to integrate expressions."
It's more or less exactly the same book I used in high school. I loved it then, but now I realise that while the order in which he taught things worked well for me, it might not work best for everyone. First he introduces the integrals of a few basic functions, and then he covers (2), (3), and (4) in that order. While he's teaching (2) - (4), he continues to introduce new basic forms that you can integrate. So he's throwing a ton of different tricks at you all at once, instead of presenting integration as a systematic process.
Here is the new approach I was thinking of using with some students:
[Okay, this approach does assume they are already comfortable with the whole "function" concept, which isn't really very difficult, as I'll explain later. Any way, my explanation in real life probably wouldn't be so heavy on function notation]
Introduce all the basic functions first, before you even teach substitution. If this bores them, spice it up with some real-world applications. By the time they get to substitution they should know all the basic forms like the backs of their hands.
(In Stewart's book, there are only 16 "basic" functions to learn, so learning them all at once isn't as hard as it sounds. And it's possible to separate some of these 16 functions into different groups, to make them all easier to remember. For example, 10 involve trig, whereas 6 do not.)
After they have been through Phase 1, they should instantly recognise all
these 16 basic forms instantly, whenever they see them.
So they'll know...
cos x --> sin x + c
x^2 --> (x^3) / 3 + c
e^x --> e^x + c
When they get to simple substitution, teach it to them in the following, very formal way:
Okay, kids, let's look at these new integrals. These things we want to integrate are not basic functions, but they all have a lot in common in terms of how they look. Each of them is the product of two different functions [I.e. in every case we have one "glob of numbers, symbols and x's" times some function which I'm going to call "f(x)." So we've always got "some glob of stuff" times f(x).]. Now that "glob" will be a composition of two more functions. [So that thing I called a glob will just be one function inside of another. Let's write this glob as "g(h(x))" ... because we can.] When you see this kind of thing, kids, "substitution" may be in play.
So at this point, we have something inside the integral that looks like g(h(x))*f(x) dx. Now ask yourself a couple questions: (i) Are g(x) and f(x) "basic functions" that we would know how to integrate? (ii) Is f(x) the derivative of h(x)? If you've answered "yes" to both questions, then substitute the new variable "u" for "h(x)", and your integrand transforms to "g(u) du."
For example, when we want to integrate [sin (x)]^3 * cos (x) dx
, we recognise that it is a "glob" times the function cos (x)
. That "glob" is the function g(x) = x^3
, composed with the function h(x) = sin (x)
. They should identify the glob [sin (x)]^3
as a composition, because visually it looks like [ ... ]^3
, with "sin(x)
" stuck inside it, and that's how compositions usually look. The student notes the x^3
and cos (x)
are basic functions that we know how to integrate; notes that cos (x)
is the derivative of sin (x)
; then substitutes "u" for sin(x)
and "du" for cos (x) dx
. They end up with u^3 du
, and they know how to integrate that.
(This is what they are taught do anyway, except that I'm making it more formal, giving names to more things, and making them conscious of every step.)
At this point in the course, take some time to revisit some of the basic forms, since some of those 16 of those could be figured from other basic forms out using the substitution method. It will reinforce the knowledge from Phase 1 they already have.
(Phase 3 & 4)
I haven't figured out the details yet on how to incorporate integration by parts and trig substitution into this formal thought process that I may end up teaching to more process-oriented kids. But I don't think doing so will be too hard. It isn't too hard for me to look at an expression and tell whether it's something that you can use integration by parts on, so the formal thought process I give to my students for analysing something and ascertaining whether it's something you can use integration by parts on won't be a long one.
Finally, I'll have to develop strategies for combining all the different techniques. There are also times when you have to tinker with the integrand a bit before it turns into something you recognise. I'd have to think up some formal way they could do that.
In summary, here's what I'm aiming for... I'm trying to take my own approach to solving these integrals, which began as a collection of unconscious gut feelings that told me what to do next while looking at a particular problem, and which became more reliable as I did problem after problem; and then turn it into some kind of conscious, formal, deliberate process that the vast majority of people -- who don't rely on gut feelings as much as James Stewart and I do when it comes to Math -- can remember and use.
Stewart doesn't give students a formal procedure for how to integrate. He shows them a few examples of how to apply the substitution rules and expects readers to figure out the procedure on their own. Then later he gives students a mess of integrals and expects them to intuitively "know" which integration tricks to use on which ones. (I'm not singling him out at all for criticism; that is how they all teach it.) I want to give students a formal, analytical way of deciding which trick works for which integral.
I'm trying to play the part of that Indiana Jones character kicking sand on the invisible bridge in The Last Crusade
. You need to make things more systematic, concrete and familiar for some people. Not everyone can blindly lunge correctly in the direction of the right answer.