So far there are four posts in this integration series:

• Integration By Parts, Part III
• Integration By Parts, Part II
• Integrations By Parts, Part I
• U-Substitution

It is good to pause here to rebalance, reorient, regroup, etc. before just marching on to the next topic.

By now, students are well-established into the following three camps:

• Completely lost
• Muddling along
• Completely comfortable

Completely Lost

The students that are completely lost are lost commonly for the following reasons:

• Some completely-lost students have been pushed through their basic math courses having memorized algorithms and gimmicks and now are faced with semi-open-ended problems that require a strong grasp of both mechanical and conceptual fundamentals and an actual ability to problem-solve. Unfortunately, these students don’t stand a chance anymore. No amount of memorization will save them. There are no helpful gimmicks. There are no shortcuts.
• Some completely-lost students have approached Calculus II the same way they’ve approached all other math classes up to this point — grubbing for points. Can I have partial credit? Extra credit? Can I write an essay? Unfortunately, integration techniques are far more mechanical than they are conceptual. The concepts are basic and simple. We are unwinding a derivative. There are some deeper concepts, but ironically enough, understanding those concepts requires some additional mathematical machinery that is out of reach.
• Some completely-lost students just have not devoted the time that they need to even though they may have the prerequisite knowledge. Integration isn’t necessarily easy stuff, but it isn’t the hardest thing in the world either. Some students just don’t respect that learning requires effort on their part.

In my experience, the students who fall under the first bullet point in the list immediately above, are beyond help. They have to actually go back and retake the elementary math courses properly. I don’t really know where to put the blame, but there’s probably some on the student, some on the teacher, and some on the general system of lower-level math education. The other two categories of completely-lost students can turn things around for themselves if they change their attitude / mindset. The teacher can step in and provide guidance on how to approach the material in an academically healthy way, but ultimately the student has to self-activate at some point.

How can you tell if a student is completely lost if you don’t already know? Or more correctly, how can you let a student self-diagnose they are completely lost? (Some students are in denial — they think that if they just stick through it, then they will amass enough points to get a minimally passing grade; not so in this class.)

Try the following problems (or variations on them):

• \(\int x^{2}\ dx\) vs \(\int x^{-2}\ dx\) vs \(\int x^{\pi}\ dx\) vs \(\int x^{\frac{3}{2}}\ dx\) — all of these problems use the same theorem: For \(n \neq -1, \int x^{n}\ dx = \frac{x^{n+1}}{n+1} + C\) where \(C\) is the constant of integration. Yet the completely-lost student may, only correctly answer \(\int x^{2}\ dx\) and likely only because the answer has been memorized.
• \(\int \cos(x)\cos(\sin(x))\ dx\) is a straight-forward \(u\)-substitution, but the presence of the trigonometric functions will cause paralysis.
• \(\int xe^{-x}\ dx\) vs \(\int \frac{x}{e^{x}}\ dx\)
• \(\int x\ln(x)\ dx\) vs \(\int x\ln(\sqrt{x})\ dx\)

Muddling Along

The students who muddle along generally are of one type (but of varying degree, if that makes sense) — they know enough of the pre-requisite elementary material (Arithmetic, Algebra, Precalculus) and have had some success with the Calculus I material, but haven’t really been able to synthesize all the concepts and mechanics as well as they should have. This type of student is at reasonably high risk of being “completely lost”; a missed class, avoiding problem sets, not reading through the book, etc. are all disengaging actions leading to complete lost-ness.

The instructor can also be an enabler of discomfort for this type of student by never giving a variety of “stepping stone” problems (some call this differentiation). Here is an example of a problem that can be enlightening for the student who is muddling along.

$$\int \sin(\sqrt{x})\ dx$$

This is also a problem that can be challenging for the student who is completely comfortable. The difference is how the problem is presented. The student who needs a bit of a helping hand can be given the above problem with a few hints. Hint 1: use \(u = \sqrt{x}\). Then let them struggle — odds are they will get to \(du = \frac{1}{2\sqrt{x}}\ dx\) and may even write
$$\int \sin(u)\cdot 2\cdot\sqrt{x}\ du$$

Then Hint 2: Why do we still have \(x\) terms if we have made a \(u\)-substitution? And hopefully, the student should be able to write
$$2\int u\sin(u)\ du$$
And from here, they can proceed with integration by parts. I prefer to give hints like this because many of these types of students are still not used to using more than one mechanic in a problem. It is a conditioning that starts at the elementary math level and is a general curse on students thereafter. This Calculus course is, in effect, the last real chance students have to recondition themselves. The rest of the undergraduate math course curriculum isn’t necessarily “hard”, but it does require students to fluidly combine elements from previous courses — and this is one of the main reasons why Mathematics appears to be hard. It’s a subject that was never meant to be learned by memorization and mnemonics but for a variety of reasons is taught that way and subsequently learned that way by many.

Anyway, I always have to have some rant in here. Back to integration and variations on a theme.

Next give students $$\int \sqrt{w}\sin(\sqrt{w})\ dw$$ and see what happens. It’s a variation on a theme, but it will require the muddling student to think and look back at what was done. Some will understand it, some will struggle and the easiest thing to do is to get students to talk about their approach.

In any case, with students caught in purgatory it is best to nudge them in the right direction, not shove them and most certainly not to ignore them (the latter is a general rule — never ignore students). Continue to give problems, continue to give hints. And if you are a student reading this, it is best to push yourself and to keep working on problems!!

Try this one $$\int e^{\sqrt{x}}\ dx$$

Another useful thing to try, though I admit that I haven’t had too many opportunities for this, is to get students to create integrals. Suppose I wanted to create a workbook of integrals for \(u\)-substitution, how would I do it? Would I really sit there thinking up of an integral, or is there a general way for me to do it? What about for integration by parts?

Completely Comfortable

It is easy to ignore the students who just “get it” immediately, but this is a teaching trap. These students can be pushed far beyond the course curriculum. The easiest way to push them is to challenge them. Challenging a student absolutely does not mean to give them 100 problems! That is another teaching trap.

Challenging a student means asking them questions from material from a section not yet covered and seeing what they can do. It means asking them to try to do the work mentally rather than on paper only (working things out in one’s mind is a really useful life skill!). It means asking them to look at $$\int \sin(\sqrt{x})\ dx$$ but providing no hints. It means taking them beyond a mechanical understanding of the material and the simple initial concepts to a deeper level of mechanics and more advanced concepts — everything in stages and in baby steps.

With that said, just because a student isn’t “completely comfortable” doesn’t mean that I hold back information. I still show all my students the same material. But as for emphasis, I try to not overwhelm or underwhelm students with material that is either beyond their reach or too simple. All students can get to a point of “completely comfortable”, it’s just a matter of time.

Here are some challenge problems for “completely comfortable” students given knowledge of \(u\)-substitutions, integration by parts, and familiarity with reduction formulas. Some of these get tricky in a hurry!

$$\int \frac{dx}{1 + \sqrt{x}}$$
$$\int \frac{1 + \sqrt{x}}{1 – \sqrt{x}}\ dx$$
$$\int \ln(x^{2}+1)\ dx$$

Also get them to see what happens if we try to “generalize”. Is this problem doable for all positive integer \(n\) or just a select few? If a select few, then which few? If all positive integer \(n\), what is the general integral?
$$\int \ln(x^{n}+1)\ dx$$

Can they write a “one-step” formula to evaluate
$$\int x^{n}e^{x}\ dx$$
rather than a reduction formula?

What if \(f\) is some even function and is well-defined on \([-\pi, \pi]\), what then is $$\int_{-\pi}^{\pi} f(x)\sin(x)\ dx$$

Since the material covered thus far is only up to integration by parts, there is only so much variety in integration problems that exist. (Well, technically, we can get as crazy as we want, but making an elaborate integral isn’t really the type of “challenge” I’m promoting.) In any case, at the end of every unit or sub-unit in the integration series, I like to ask students to try and tackle:

$$\int \sqrt{x^{2} + 1}\ dx$$

I do this because I want them to get a feel for when integration techniques don’t work and I also want them to start to develop a “hierarchy” of integration techniques. \(u\)-substitution undoes the chain rule. Integration by parts undoes the product rule.

For the above problem, \(u\)-substitution is immediately out, but students will try this anyway. Then they are left with integration by parts because that is all they currently know and after a few fits and fights they may end up with $$\int \sqrt{x^{2} + 1}\ dx = x\sqrt{x^{2}+1} – \int \frac{x^{2}}{\sqrt{x^{2}+1}}\ dx$$ and the integral on the right isn’t really all that appealing either. But it looks similar to the integral on the left and out of an overfamiliarity with some other techniques (hey, overfamiliarity is at least better than no familiarity!) they may end up with
$$\int \frac{2x^{2}+1}{\sqrt{x^{2}+1}}\ dx = x\sqrt{x^{2}+1}$$

which hasn’t really helped them in their original question. But this is a valuable lesson regardless and it is a mild hook into integration via trig substitution and integrals involving powers of sine and cosine. And conveniently that’ll be the next topic.

What types of challenge integrals do you give your students?