How Much Mathematics Should a Student Memorize? Part 2, Integral Calculus

My basic attitude towards memorization in mathematics education is to memorize the absolute minimum, but memorize that minimum perfectly. Part of a mathematics teacher’s job, in my view, is to guide students to understand what this “minimum” is, and then encourage them to memorize it, helping them to find effective means for memorization.

Effective means, in my experience, typically involve associating the fact to be memorized with a diagram, graph, or other visual representation, or using the fact repeatedly in the context of solving problems. Just staring at a fact and trying to force it into one’s mind isn’t nearly as effective, in my experience. But everyone is different, so each student must experiment with what works best.

The advice to practice facts that you wish to memorize in the context of solving problems dovetails with another slogan, which I explored in a previous post:

The more you understand, the less you have to memorize.

And now for the subject of this post.

Many years ago, when I taught calculus for the first time, I was preparing a lecture on integration of trigonometric functions. When I came to the problem of teaching students the following integral

$\int \sec x \, {\rm d}x$

I was unhappy with what my textbook contained by way of explanation. The current textbooks are no different … stuff tends to get copied from one book to the next, doesn’t it? Here’s what one textbook on my shelf says:

“We will also need the indefinite integral of secant.” The result is quoted, and then: “We could verify [the result] by differentiating the right side, or as follows. First we multiply the numerator and denominator by $\sec x + \tan x$:”

$\int \sec x \, {\rm d}x = \int \sec x \dfrac{\sec x + \tan x}{\sec x + \tan x} \, {\rm d}x$

$\int \sec x \, {\rm d}x = \int \dfrac{\sec^2 x + \sec x \tan x}{\sec x + \tan x} \, {\rm d}x$

The textbook goes on to say that by substituting $u = \sec x + \tan x$, the result follows.

This left me (and still does leave me) profoundly unsatisfied. As George Polya advocated, the role of a mathematics teacher is to guide students to solve problems for themselves, or at least see how they might have solved them. Pulling this particular rabbit out of this particular hat does not lend itself to solving this integral by oneself. How on earth would anyone but a genius come up with this trick?

So I set out to see if I could integrate secant in a straightforward way. One strategy I teach is that if you are faced with a trigonometric integral, and can’t see anything better to do, convert the expression to a combination of sines and cosines. OK, let’s try:

$\int \sec x \, {\rm d}x = \int \dfrac{1}{\cos x} \, {\rm d}x$

If we are thinking about making a substitution, we need some expression in the numerator that is likely to be a little more complex than 1. So let’s multiply the numerator and denominator by something. What? Well, we are trying to work with just sines and cosines, so let’s multiply the numerator and denominator by either $\sin x$ or $\cos x$. The former doesn’t seem to lead anywhere, but the latter does:

$\int \sec x \, {\rm d}x = \int \dfrac{\cos x}{\cos^2 x} \, {\rm d}x$

Bingo! Look at the denominator. It can be easily converted to sines:

$\int \sec x \, {\rm d}x = \int \dfrac{\cos x}{1 – \sin^2 x} \, {\rm d}x$

Then the substitution $u = \sin x$ is very natural, because the derivative of sine is sitting right there in the numerator. After the substitution, we are left with

$\int \sec x \, {\rm d}x = \int \dfrac{1}{1 – u^2} \, {\rm d}u$

Now we have an easily factored expression in the numerator; just use the method of partial fractions, which results in two easy integrals, then use properties of logarithms to recombine the expressions, do some algebra, and Voila! The result from the textbook is obtained.

I actually showed my class (all those many years ago) this long calculation, and was met with the question, “Do we need to memorize this proof?” My answer was that because this calculation is so long (and therefore time-consuming on a test or exam), because it is such a shaggy-dog story of integration, that it is essential to memorize the integral of secant.

As a student, I never bothered to memorize the integral of tangent, because I had practiced the idea of switching to sines and cosines, making a simple substitution, and doing a couple of lines of algebra. I practiced this often, and soon I could do the calculation in seconds. Of course, at some point the fact became embedded in my head, but it was memorized in a very natural way, without having to devote effort to it. And this is part of my point in the “memorize the minimum” slogan. I would rather have students practice integration problems than waste time memorizing things that will soon be forgotten if they are not continually used.

Of course, nowadays we have software readily available, and so the whole question of how much technique of integration one ought to teach students is in the air. My view is that some of it is still useful, in the same way that learning various algorithms (say, long division) is still important, even though calculators can divide in an instant. It is only by getting one’s hands dirty with specific calculations, and solving specific problems, that one can be an intelligent user of mathematical technology. But that is perhaps a discussion for another time.

I was interested to find out how today’s textbooks handle teaching the integral of secant, so I quickly checked what is available on my book shelf. I checked Stewart; Anton, Bivens, and Davis; Briggs and Cochrane; Smith and Minton, Thomas and Finney (an older book); Hass, Weir, and Thomas; Rogawski; Adams and Essex; Edwards and Penney. All of them do exactly the same thing that the textbook I used so long ago did, which is to multiply numerator and denominator by secant + tangent.  One of them had the grace to call it “an unmotivated trick,” another called it a “clever substitution.”

I call it pedagogically unsatisfactory.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)