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

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?

• When I took Calculus I always used $u$-substitution where applicable. It prevented me from making any careless mistakes when finding the antiderivative. In Calculus 2 my TA would skip the $u$-substitution and when going over a problem, probably due to having more experience. There is no harm in teaching it and letting the students decide what works for them. I think that students who are confident enough can eventually skip $u$-substitution, while other students may find it to be a useful tool. Dec 19, 2020 at 3:11
• Your question is not far from slightly more explicit, $\int \cos^3(x)\sin(x) dx = -\int \cos^3(x) d(\cos (x)) = -\frac{1}{4}\cos^4(x) + c$. This also avoids the explicit label for the substitution, but makes more obvious the substituted variable. More to the point, when we ask questions which permit guessing, it's a little unfriendly to take off points for not showing work. Dec 19, 2020 at 19:02
• If you want some extreme examples, you could flick through the Integration tag on Math.SE. A great many of the top ones have some (often several) clever substitutions in. Jan 2, 2021 at 13:01
• @JamesS.Cook This notation, which I've seen only on mathematics.SE, has been puzzling me until your above comment (where did it originate?)! So, $$\int_a^b f\big(g(x)\big)\, g'(x)\,\mathrm dx :=\int_{g(a)}^{g(b)} f\big(g(x)\big)\,\mathrm d\big(g(x)\big) \ ?\tag1$$ Feb 9 at 4:24
• @ryang my wife was taught calculus in Hong Kong and I think I saw the notation with some of her curriculum, I've also seen it in the work of some rather advanced Asian students who have crossed my path. I do like the reference $x$ notation for the bounds, that is a nice sort of middle ground. Of course there exist many applied problems where changing the bounds is actually critical to understanding the formulas derived. For instance, separation of variables to derive the timeless equation in kinematics. You want position and velocity as bounds, so changing bounds is physically nontrivial. Feb 9 at 5:30

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.

• Great answer, especially the point about geometry underlying certain substitutions. I would add, I prefer the terms "explicit substitution" in place of u-substitution since that is what it is, whereas I prefer the term "implicit substitution" in place of trig-substitution. Dec 19, 2020 at 18:56
• Exactly. To go further, I recommend explicitly teaching students how to use the word "let" to invent variables as placeholders for inconvenient quantities. It's a very general tactic, as this answer explains well. Dec 19, 2020 at 21:40
• My personal extra round of applause for the last paragraph! Calling this method a "$u$-substitution", which seems to be specific to math in the English-speaking world, is such a terrible convention. I try to explicitly break it every time I teach Calculus. First, I literally tell my students that this is a terrible name and we must call this method a "substitution". Second, when going over examples I use a variety of letters for my substitution. I don't think I'm being super-successful on my crusade, but at least I'm trying. :-) Dec 24, 2020 at 21:10

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.

• "There are sometimes reasons to avoiding" - to save time, of course. I'd say that we need to introduce the method, and let the more advanced students know when they 'see' the process easily, they can skip the actual use. I just solved an equation with radicals on both sides. 12 steps. And told the class that with practice it could be 2-3, by combining and/or skipping steps. Knowing the very long way is a great start. Dec 19, 2020 at 16:49

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.