Integration by Substitution

Introduction to this topicEdit

This page is dedicated to teaching techniques for integration by substitution. For other integration methods, see other sources.

The first section introduces the theory. Next comes a demonstration of the technique; this is followed by a section listing the steps used in that demonstration. The last section is a series of clarifying examples.

Assumed KnowledgeEdit

  • Basic differentiation
  • Basic integration methods

To understand the theory:

  • Function composition: that a function   can also be written as  , which we call the composition of   and  .
  • Chain rule

Theory of Integration by SubstitutionEdit

This area is covered by the Wikipedia article Integration by substitution. On this page we deal with the practical aspects.

We begin with the following as is described by the Wikipedia article

 

This can be rewritten as

 

by setting

 

The principle applied here is function of a function (Function composition) and the reverse of the chain rule. This is the basis of integration by substitution.
The key skill now is to identify what value we use for   and following the process to solution.

Integration by substitutionEdit

The objective of Integration by substitution is to substitute the integrand from an expression with variable   to an expression with variable   where  

Theory

We want to transform the Integral from a function of   to a function of  

 


Starting with

 



Steps

          (1) ie    
       (2) ie    
       (3) ie    
       (4) ie   Now equate   with  
       (5) ie    
       (6) ie    
       (7) ie   We have achieved our desired result

Procedure

  • Calculate  
  • Calculate   which is   and make sure you express the result in terms of the variable  
  • Calculate  
  • Calculate  

TechniqueEdit

Example 1Edit

Let us examine this integral

 

The inner function is

 

The outer function is

 

Recognising this relationship we then move onto the following set of steps to process the inner function
NOTE: that the differential of   is  .

 

Now we substitute   and   into the original integral.

 

Then apply standard integral technique

 

And finally we substitute the value of   back into the equation

 

Example 2Edit

Let us examine this integral

 

We can first rearrange the fraction to make it more familiar.

 

The inner function is

 

The outer function is

 

Next we assign   and  

 

But we have a problem!   doesnt equal  ! So we need to rearrange our formula for  .

 

Now we can substitute   and   into the original integral.

 

Study the above substitution carefully. We moved the fractional component of du to the front as it represents a constant.

Now apply standard integral technique

 

Cleaning up this expression we have

 

And finally we substitute the value of u back into the equation

 

The Definite IntegralEdit

Consider the definite integral

 

By using the substitution

 

Now because we have limits, we need to change them with respect to  . Note the value of the limits.

 
 

Now we have a new definite integral to solve

 

The Steps We AppliedEdit

Let's now review the steps for integration by substitution.

Indefinite Integral Definite Integral
1. First identify that you have a function of a function. This skill comes with practice to identify candidates. First identify that you have a function of a function. This skill comes with practice to identify candidates.
2. Identify   and then find   that is appropriate for the expression. Identify   and then find   that is appropriate for the expression.
3. Change limits for definite integrals.
4. Integrate using normal techniques. Integrate using normal techniques.
5. Substitute back the values for u for indefinite integrals.
6. Don't forget the constant of integration for indefinite integrals.

Finding uEdit

Let's look at more examples at finding  .

Example 1Edit

 


       
       
     
     
 

Example 2Edit

 

Here we first perform the substitution  , so that

 

With this, we get