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Calculus Substitution Rule Alternative Method

  1. Apr 10, 2015 #1

    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.
    Last edited: Apr 10, 2015
  2. jcsd
  3. Apr 10, 2015 #2


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    Homework Helper

    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.
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