To determine if a ‘u substitution’ is appropriate for an integral, look for the value of du somewhere in the original integral. You may need to multiply or divide by a constant to get du in the integral.

Example 1: Integrate:

Here the obvious choice for . And the derivative of so is in the integral too. Make the substitutions and solve:

Example 2: Integrate:

Do you see that is the derivative of if we multiply by Therefore, let and To get the factor of in the integral, multiply by and multiply on the outside by The problem then becomes:

Example 3: Integrate:

How is this a u substitution problem? First, rewrite

Now do you see the u sub? Let Multiply inside and outside the integral by so you can make the following substitution and solve:

What if you let in this example? Then This seems like a reasonable substitution. However, this will put in the denominator. And that’s a definite no no! Any time you make a u substitution, the term must be in the numerator.

Note: There are a couple of instances where a u substitution works and du isn’t in the original integral. You will learn these with a little practice. Here’s one example. Note that the radical term is not the derivative of the x term and the x term is not the derivative of the radical. However, with a little rearranging, a u substitution leads to an expression you can integrate:

Substituting gives