Calculus Substitution Rule Alternative Method

[eurythmistan]

Hi,

I'm not sure if this should actually be in the "homework" section instead. I'm posting it here because it's more of a pedagogy question, I think, but I could be wrong about that also.

Ok, I tutor calculus, and when I do u-substitution, I always solve for something (not always dx), so that when I replace the variables, all the x's get replaced at once. There is no "mixing" of x's and u's in the same integral. However, I've noticed some of my students use a method whereby they always solve for dx, and then replace just some of the x's, have some u's mixed in, and then cancel out the rest of the x's for the next step. This to me just looks wrong, and contradicts most if not all the references I've seen on the topic. It's also not the way I'm inclined to work through the problem naturally.

I realize that one should get the same answer either way, so is there any good reason to tell them that I think they should do it my way, other than that I think they'd be able to follow the examples in a textbook better?

ETA: I realize this may be a non-issue, so feel free to give that answer, too.

 

[RUber]

I think that mixing x and u terms is a product of being told what u to use in the substitution. In practice, if you don't know which u to use, you are looking for one which will let you go straight to the finish line. For working through the calculations, I don't see any harm in mixing the terms as long as none are leftover after the substitution is complete...it is just showing the work. However, to justify why students should get into the habit of dropping the intermediate step--I would say it will help them to build intuition which will be necessary later on.

 

[martane]

Hi there,

As a fellow calculus tutor, I can understand your concern about the different methods used by your students for u-substitution. In my experience, there is no one "right" way to do u-substitution and ultimately, it should come down to what works best for each individual student. Some students may find it easier to solve for dx and then replace variables, while others may prefer solving for something else and replacing all the x's at once.

The important thing is that they understand the concept of u-substitution and are able to apply it correctly to solve the integral. As long as they are getting the correct answer, I don't think it's necessary to insist on a specific method. However, it may be helpful to explain to your students the reasoning behind your preferred method and how it can make the process more efficient and less prone to errors.

Ultimately, the goal of tutoring is to help students understand the material and find the approach that works best for them. If your students are able to successfully apply u-substitution, then I wouldn't worry too much about the method they use. However, if you feel that their method is causing confusion or errors, it may be worth discussing it with them and offering alternative approaches. Hope this helps!