When g'(x) = f(x), you can use the substitution u = g(x) to integrate expressions of the form f(x) multiplied by g(x). Variable substitution helps to fill the gaps left by the absence of a Product Rule and a Chain Rule for integration.

Some products of functions yield quite well to variable substitution. Look for expressions of the form f(x) multiplied by g(x) where

• You know how to integrate g(x).

• The function f(x) is the derivative of g(x).

For example:

The main thing to notice here is that the derivative of tan x is sec2 x. This is a great opportunity to use variable substitution:

1. Declare u and substitute it into the integral:

2. Differentiate as planned:

3. Perform another substitution:

4. This integration couldn’t be much easier:

5. Substitute back tan x for u: