Substituting with Expressions of the Form f(x) Multiplied by h(g(x))
How to Integrate Rational Expressions Using the Sum, Constant Multiple, and Power Rules
Integrating Powers of Cotangents and Cosecants

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:

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
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Computing Integrals and Representing Integrals as Functions
Find the Integral of Nested Functions
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