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dethwing · DeTheeThaw

So a few nights ago, I was tutoring a calculus student in a u-substitution problem. I won't go into all the details, but basically the part I want to ask about was this.

du = 2xdx

And there was an "x dx" in the problem. In order to do this, do you:

1. Get du/2 = xdx , and then replace both at once

2. Get du/(2x) = dx, and then replace just the dx, canceling the x/x.

Now, I was taught to do it the first way, so I got thrown off a bit when my student starts dividing by x, but then I saw what was going to happen and let it go. So I was just curious, is this a newer way of doing it? Is it just a matter of the text/teacher?

Zag · Unintentionally offensive old coot

Neither approach is wrong. One is a little cleverer and faster, but a fundamental approach of wanting to remove the dx from the other problem, so solve for it in this one and substitute is always going to yield a correct result. If you happen to notice that there is more in the original problem you can substitute for in one step, that's fine, too; but one might waste more time looking for that sort of thing than benefit from the times finding it.

Of course, when the x is something more complex, that you have to take some pains to construct in order to make the substitution work cleanly, it's nicer to be more ready to use your approach. On the other hand, it is never strictly necessary.

Trojan Horse · Daedalian Member

As Zag said, both methods are correct. I greatly prefer the first one, though.

I don't want my students to get into the habit of replacing dx with a "mixed expression" like du/2x. What happens when they come across a problem where a u-substitution does not work, such as integrating dx/(x^2+4x+3)? I fear they would try the substitution, getting an expression involving both x and u... and then they'd shrug, and just keep on going with the problem.

No matter how many times I tell my students, "you have to rewrite the entire integral in terms of u and du, with no x's left over, or you can't do the substitution", some of them still forget. And I think things would be worse if I regularly did things like substituting du/2x for dx.

Lepton* · Guest

Agree with both. The first approach is the one we use once we know what we're doing, so that's what I teach.