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.