Integrating with u-substitution – Knowing when to use u-substitution

Two common questions when learning u-substitution method of integration is “how do I know when to use u-substitution” and “how do I know what to choose for u?”  A large part of the answer is simple: “experience”.  But one key concept that students often miss is that the process of u-substitution depends heavily on knowing your integration rules.

For the Calculus I class, students should always be thinking about their known integration rules.  These are the rules that are often listed on the inside front or back cover of a calculus book, or in an appendix.  The point of knowing these integration rules is that they are the ONLY integrations you can perform.  What I mean by this the following..   Suppose that you were given a problem to solve:

The beginning Calculus I student should say to himself “Hmm… there is no integration for integrating 2x-squared.  Therefore, I need to alter the problem.”  How the student alters it is not critical (as in math there is often more than one way to solve a problem), but the end result is that the problem must turned into a known integration rule.  In this case, one of the known integration rules is the general power rule:

So the first question then becomes “how can I modify the first problem so that it looks like the power rule?”  If you always remember that you can only integrate from a known rule, then you’ll have a much easier time knowing when and how to use u-substitution.

In this case, there are two easy paths to a solution. 

Option 1: Perform the algebraic square operation first, and then perform the integration using power rule		Option 2: Use u-substitution

In both cases, we modified the integration problem that was not a known rule to a version that was a known rule.  In option 1, the algebraic square made the problem a known rule – the power rule (the fact that it is ‘x’ instead of ‘u’ is of no consequence.)  In option 2, we directly transformed the original problem into the power rule.

What makes this easier is that in Calculus I, there aren’t very many “known rules”.  There is the power rule, trig rules, logarithmic rules, etc.  You also have other techniques that are available (integration by parts, for example) but even those other techniques provide known integration rules for you to use.

So the key point is that you can only perform integrations using the known rules, or else you have to modify the problem at hand to use a known rule.

The second question is “once I choose a ‘u’, can I get a ‘du’ also?”  Unfortunately it isn’t enough to just assume that du = dx and that you can replace one for the other.  You actually have to solve for what du is in terms of dx, and then make appropriate substitutions in the problem.  There are also two ways to think about this process.  Using the same problem as above, we could say:

Option 1: Solve for any remaining terms in the integration problem after you’ve replaced the ‘u’ (in this case, just ‘dx’)		Option 2: Alter the original problem to make the terms match, while making sure you don’t change the problem

Personally I find option #2 more straightforward, but you have to be careful because you can really only use constants (not variables) in your manipulations.  For example, let’s say you needed ‘2x dx’ instead of just ‘2 dx’.  You can’t multiply the internal equation by x and then put 1/x to cancel it out.