A u-substitution question

In a u-substitution integral, like this one:

…one chooses a value of u from within the integrand and then computes du. For example:
Then you have to substitute stuff into the integral to make it computable. Here, you’d multiply inside the integral by 2 and outside by 1/2, allowing you to make substitutions within the integrand and then get a pretty easy antiderivative.

Could somebody give me an explanation of exactly why it’s wrong to do the following instead of multiplying on the inside by 2 and on the outside by 1/2:
(1) Solve the second equation above for dx (here, you’d get du/6x)
(2) Plug the resulting expression back into the integral for dx, and try to cancel stuff out (here, you’d cancel out the x and would be left with a constant factor of 1/2).

What I’ve been telling my students is that (a) it’s just a bad idea to mix variables in general, and (2) the du and dx terms are not to be manipulated at will like the numerator and denominator of a regular fraction. But I’m not sure they’re convinced, and neither am I frankly. My colleagues view the mixing of du and 6x as anathema. But I would like to have something more substantive than a sort of aesthetic argument.

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2 responses to “A u-substitution question

  1. I have always used the dx=du/6x approach to replace as many of the x’s by u’s as possible and then decide what to do if the x’s don’t cancel. In all the years that I’ve taught this way not one student has asked the awkward question about what happens if x is zero 🙂

    My reason for doing this is to concentrate on actually doing the integration without letting the technicalities get in the way. I would expect that in a later course a more rigorous method could be used, but such rigour is more easily understood after being able to do the integration. At first contact I would be very reluctant to use the formula here as I think understanding comes after doing in much of mathematics (though I wouldn’t take it to ridiculous extremes).

    I do use the trick of multiplying and dividing by a constant with certain fractions so that I can use the “when the the top is the derivative of the bottom, the integral is the log of the bottom” rule, because I find that students almost always forget to divide by 2 when integrating 1/(2x+1) for example.

    Which reminds me of this joke.

  2. Just another liberal professor

    Honestly, there’s nothing terribly wrong with it…if you treat du and dx as differentials. Personally, I go with the approach of saying that x dx = (1/6) du and (if the students are fairly new to stubsitution) circle the “x” and “dx” and put arrows from them to the “du” in the transformed integral. Of all the approaches I’ve used, that one seems to work the best, although some students still like to say that dx = du/(6x) and go from there. Although Steve does bring up an interesting point about what happens if x=0, so I guess saying that dx=du/(6x) might not be 100% kosher…I guess I can add that to the list of “lies we tell our calculus students”.