Method of Substitution

Abstract

Opposite of the chain rule

  • Set the inner function to variable, usually , and differentiate it to get a differential with (e.g )
  • You need the match the with the in the integrand
    • If this is already in the integrand, it gets "absorbed" into the
    • Otherwise, you have to create if yourself with
  • Then, sub in the and , and then take the integral with
  • Plug back in when finished integrating

The Method of Substitution (AKA the Change of Variable Technique)

Recall the chain rule:

Which means its antiderivative is

If we notice and , then we can set , so , and in differential form, this will be . This allows us to write out integral in terms of a new variable:

Then we just use the integration rules

Definition

where

Example

Evaluate

solution
Let , which means

Example

Evaluate

solution

Let

Example

Evaluate

solution

Let

Example

Evaluate

solution

Let

Example

Evaluate

solution

Let
Also,

Example

Evaluate

solution

Let

There are many forms to this answer.

Example - Definite Integral

Evaluate

solution
Let
Note that when changes from to , increases from to .

Example - Definite Integral

Evaluate

solution
Let
Note that when changes from to , changes from to , which is from to .