Substitutions in Calculus - U Can Do It (Maybe Not)

With the AP Exams this week and the year coming to an end, the timing of this post is way off but I wanted to share some thoughts about using u-substitution in integration (and differentiation). Every text promotes it, I've been teaching it for years, but, like so much else in mathematics, u-substitution is really a form of 'information-hiding' that may sacrifice understanding for the sake of brevity. While I believe using u in place of f(x) is an important technique for some calculus algorithms, experience has shown me that keeping f(x) in the formulas works better during the learning phase. In this post I will show how using f(x) in the formulas for integrals may help students deal with the variety of functions they encounter when integrating using the 'Inverse Chain Rule' (I coined that phrase to help students recognize that the f'(x) factor disappears when anti-differentiating (integrating) and appears when differentiating. If your first reaction is that these formulas are more complicated, I think you might need to actually experience the difference in students' reactions. There's no substitute for that!!

If you've already tried both approaches, let me know which form worked better for a majority of your students. My impression is that many students are at first overwhelmed by the varieties of different forms that appear in the typical set of exercises following this lesson. We can say that they're basically all the same, but it doesn't appear that way to the novice!

Outline of Lesson:
The topic was introduced by reviewing several differentiation problems using the General Power Rule for d/dx(f(x)), using the Chain Rule. Each derivative was immediately followed up with the corresponding antiderivative problem. The idea that both (f(x))ⁿ and f'(x) must be present in the integrand was emphatically stressed. I repeatedly asked the question, "What is the f(x) here?" then, "What would f'(x) be?" Also, "Is the exact f'(x) in the integrand?" If not, we discussed the method of supplying a missing numerical factor and its reciprocal, using properties of antiderivatives, etc. They seemed to get this, struggling a bit with the idea of using differentiation in an integration problem, a natural source of confusion at first. We made sense of all this for the first few examples by differentiating our answer to check it. As with all new ideas, some caught on quickly, others had not yet internalized it after 3-4 examples because I too quickly moved into variations on the basic form before they had really processed it.

The following are the 'forms' that helped them adjust to the variety of trigonometric and other types of examples. This seemed to eliminate much confusion...

Some ∫ form(ula)s:

1. (General Power Rule): ∫f'(x)(f(x))ⁿdx = (f(x))ⁿ⁺¹ /(n+1) + C,
n ≠ -1

2. ∫cos(x)sinⁿ(x)dx = (sinⁿ⁺¹(x))/(n+1) + C, n≠ -1

3. ∫f'(x)sin(f(x))dx = -cos(f(x)) + C
OR
4. ∫f'(x)cos(f(x))dx = sin(f(x)) + C
Note: There is a critical but subtle difference between #2 and #3, even though they are both applications of the Chain Rule.

5. ∫f'(x)sec²(f(x))dx = tan(f(x)) + C
6. ∫f'(x)sec(f(x))tan(f(x))dx = sec(f(x)) + C

7. ∫f'(x)e^f(x)dx = e^f(x) + C

8. ∫f'(x)/f(x)dx = ln|f(x)| + C vs. ∫du/u = ln|u| + C

There are many others, but you get the idea.
If you still believe that the u-forms are so much simpler, that's fine, since I like the u-method as well and it is needed later on. I think all experienced math teachers latch on to their favorite methods or models because they have worked effectively with many students. However, I do not believe that any method will be effective for all due to learning style issues. There may be some students who will simply feel more comfortable and perform better if they use Method A vs. Method B.

I should have included some of the actual examples used, but...