Usefulness of $u$-substitution in and beyond early Calculus?

Question

My students, when presented with an integral (source) like $$\int (2x+2)e^{x^2+2x+3} \ dx$$ are apt to recognize derivative patterns like $u' e^{u}$ and reverse-engineer anti-derivatives rather than to utilize $u$-substitution.

With something like $$\int x \cdot \cos(5x^2) \ dx$$ they would rewrite to get $$ \frac{1}{10}\int 10x \cdot \cos(5x^2) \ dx$$ and then unwind the chain-rule.

So I wonder if they are missing out by avoiding $u$-substitution. Are there applications later on that will be more challenging if they don't get more practice with it now, or is $u$-substitution occasionally convenient but unnecessary?

Answer 1

A substitution is a very general procedure in mathematics. Your students will have experienced it many times already in algebra before they got to calculus, e.g., when solving two equations in two unknowns. More generally, it's an example of a change of variables, in the case where there is only one variable. An example in two variables would be changing from coordinates $(x,y)$ to new coordinates $(u,v)=(y,-x)$, i.e., a 90-degree rotation. As a fancy example from an advanced subject, you could look at the transformation from Schwarzschild coordinates to Kruskal-Szekeres coordinates for a black hole.

It's often convenient to do a change of variables simply for convenience of notation. For example, if I was going to do a lengthy calculation involving a cylindrical tank, I might find it useful to switch from the radius to the cross-sectional area, just to keep the writing simpler.

So the attitude should be simply that substitutions are things that we do naturally whenever it's convenient. They make life better by keeping things simpler. Who wants to make things hard?

Even within the context of freshman calculus, there are plenty of cases where your students' pattern-recognition strategy won't work, e.g., $\int dx/\sqrt{1-x^2}$.

As an example where you really, really want to do some substitutions, consider the problem of finding the magnetic field inside a solenoid of finite length, by integrating over loops of current. This gives an integral of the form $B=\int_p^q a^2(a^2+z^2)^{-3/2} dz$. Students should be taught that the very first thing to do here is the substitution $u=z/a$, simply because it cleans up the integral. After that, the substitution $\tan\theta=1/u$ makes the integral into a trivial one. Moreover, the final result is simpler when expressed in terms of $\theta$ than in terms of $z$, so it should just be left that way. And $\theta$ also has a direct geometrical interpretation.

A similar example would be something like $\int_{-\infty}^\infty a^2(a^2+z^2)^{-3/2} dz$. It should be explained to students that this is an indefinite integral, not a definite integral, and that the first step should be to make it into a definite integral with the substitution $u=z/a$. In real-life applications, this type of substitution is almost always obvious because we want to change from a variable that has units to one that is unitless.

IMO the term "$u$-substitution" should be dropped. We should just call this a substitution or a change of variables. That would make it more clear that this is just a normal and familiar mathematical practice. Furthermore, it would disabuse students of the notion that they will never have to make up their own notation for a change of variables -- something they are incredibly reluctant to do, because in general, they are almost never asked in a math course to make up notation for something.

Answer 2

There are sometimes reasons to avoid $u$-substitution if you can see the antiderivative without it: in a definite integral, you don't have to reparameterize the interval's endpoints to $u$-values/units if you don't use $u$-substitution. Of course, this can also be a downside in some circumstances.

The fundamental thing is, though, that $u$-substitution is really a reparameterization of the function and is intimately tied to the chain rule by the fact that we do reparameterization through composition.

Answer 3

Differentiation is a science; integration is an art.

When you integrate, you have to guess what function differentiated to give the one you are looking at. Substitution is one of the methods available for re-writing the function in the hope of finding something more recognisable.

Students who skip the substitution are actually showing more understanding than those who don't. It shows that they understand the chain rule well enough to recognise its effects without needing to explicitly write the substitution out (which just takes up more time). Doing this should be encouraged.

You could argue that they are missing out on practice of carrying out the substitution method that they might need with harder cases. If the students are able to read and understand the mathematical notation properly, they won't need so much routine practice to be able to carry out the mechanical part. What they do need is practice at identifying situations where a substitution is useful (ie. where it's much harder to spot the anti-derivative). If they are not getting this practice it is because the questions they have been set are not suitable.