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

By Mark Zegarelli

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:

image0.png

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:

    image1.png

  2. Differentiate as planned:

    image2.png

  3. Perform another substitution:

    image3.png

  4. This integration couldn’t be much easier:

    image4.png

  5. Substitute back tan x for u:

    image5.png