Using the Product Rule to Integrate the Product of Two Functions
How to Integrate Even Powers of Secants with Tangents
How to Integrate Odd Powers of Sines and Cosines

Substituting with Expressions of the Form f(x) Multiplied by g(x)

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:

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