So you’re teaching a Calculus course and you’ve gotten to \(u\)-substitution. Eventually, we get to a starter problem like this
$$\int_{0}^{1}e^{2x-8}~\ dx$$
Now, it’s possible to integrate this by inspection, but to demonstrate the \(u\)-substitution technique, we go ahead with the choice of \(u = 2x – 8\) yielding \(du = 2 ~\ dx\). Then, the rearrangements and substitutions come into play and different instructors have different styles. One way is to rewrite the integrand as
$$\frac{1}{2}\int_{0}^{1}e^{2x-8} \cdot 2 \cdot dx$$
This allows us to show that \(du = 2 ~\ dx\) is substitutable by \(du\) provided that we introduce a \(\frac{2}{2}\) term. Another approach is to “solve” for \(dx\) as \(\frac{du}{2} = dx\) and proceed with the substitution. Regardless, we are left with
$$\frac{1}{2}\int e^{u} ~\ du$$
and then it’s a matter of working out the bounds of integration. Again, there are different styles and situational needs. One approach is to proceed with integrating the indefinite integral given above, then unwinding the substitution with \(2x – 8\) everywhere there is a \(u\), and then evaluating the integral at its original bounds.
Another approach is to immediately change the bounds of integration and work with the definite integral
$$\frac{1}{2}\int_{-8}^{-6} e^{u} ~\ du$$
Here the change of bounds can be found by using the relationship chosen, namely \(u = 2x – 8\).
Both approaches are clean and are widely practiced. I have done this, as well.
However, over the years, I’ve changed my approach. The \(u\)-substitution part is fine. But where things start to get a little opaque is with respect to changing the bounds of integration. Students tend to get confused about when to change the bounds and when not to change the bounds … and why to change the bounds.
What I have started to do in the Calculus sequence is to hammer home the point that the bounds of integration are related to the variable of integration. That is, instead of writing
$$\frac{1}{2}\int e^{u} ~\ du$$
I will write
$$\frac{1}{2}\int_{x = 0}^{x = 1} e^{u} ~\ du$$
as an explicit reminder that while we made a \(u\)-substitution for the integrand, our bounds of integration are still in terms of \(x\). Thus, we can continue with finding an anti-derivative of \(e^{u}\) with respect to \(u\), but when it comes time to evaluate the integral, we can decide if we’d rather change the bounds so that they are in terms of \(u\) or write \(u\) in terms of \(x\) leaving the bounds unchanged.
Students I’ve worked with have found this explicitness helpful and clarifying. It demystifies the process. While it is true that Calculus texts do explain the variable associated with the bounds of integration, the notation becomes simplified too early for students to grasp what is going on. This confusion is similar to the confusion that students have with identifying the coefficient of \(x\) when the \(1\) is hidden.
Thus, I have now preferred a little extra writing before going towards the standard shorthand. From here, it becomes more easy to have a conversation about bounds of integration written explicitly as non-constant functions like those seen when exploring volumes of solids. Then, it’s not so strange to see \(f(x)\) and \(g(x)\) for integration bounds; and if we want, we can simply begin writing \(u(x)\) rather than \(u\) to further explain that the bounds of integration (in the example above) have really just gone from \(x = 0\) and \(x = 1\) to \(u(0)\) and \(u(1)\).
Generally and broadly speaking, when syntactic shortcuts come too early in the learning stage, further learning becomes more difficult because the details are literally missing.
This is what I have always done. And always taught in my Calculus dabbles. It is so clean.
For examples like the trick question in your comments, I would probably go ahead and change the hounds after getting stuck,but I might honestly simply realize it had to be zero because 2x e^x^4 is clearly odd. And I’d encourage students to at least this about that.
Yes, agreed. And to Joseph’s point, broadly speaking, certain pathways are more safe than others.
I do my best to convince my students to immediately change the bounds on x to bounds on u. To try to convince them, I do an example like this in class:
Integrate 2x*e^(x^4) from x=-2 to x=2.
When you do the substitution u=x^2, you end up with the integral of e^(u^2). Which means you’re sunk… unless you realize that the new bounds on the integral are u=4 and u=4, which means that the integral equals 0.
An example like this usually converts most of the class, but not all. To convert the rest of the class, I usually put an example like this on the exam, heehee.
Yes, absolutely! Part of the art form is recognizing such matters.
I do this sometimes, too. The change of bounds thing throws everyone, and the fact that there are two ways to approach it (at the beginning or do it all at the end) makes it worse.
Hard to commit to doing it all the time, though… writing that “x=” is going to be a tough habit to adopt.