Trigonometric Substitutions

Introduction to this topicEdit

This page is dedicated to teaching problem solving techniques, specifically for trigonometric substitution. For other integration methods see other sources.

The format is aimed at first introducing the theory, the techniques, the steps and finally a series of examples which will make you further skilled.

Assumed KnowledgeEdit

  • Basic Differentiation
  • Basic Integration Methods
  • Pythagoras Theorem

Theory of Trigonometric SubstitutionsEdit

This area is covered by the wikipedia article W:Trigonometric substitution and the wikibooks module B:Calculus/Integration techniques/Trigonometric Substitution. On this page we deal with the practical aspects.
We begin with the following as is described by the above sources.

Trigonometric substitution is a special case of simplifying an intergrand which has a specific form. We will first outline these forms and where they came from.

Pythagoras TheoremEdit

We should be familiar with pythagoras theorem for a right angled triangle.

 

From this familiar definition we can derive other definitions. eg.

 

By expanding upon this theory we can come up with other relationships which help us with integration.

Definition 1 Sine Substitution - containing a2x2Edit

 

From the diagram
 
 

 
 

 

 

 

 

 

 

 

Definition 2 Tan Substitution - containing a2 + x2Edit

 

From the diagram
 
 

 

 

 

 

 

 

 

 

Definition 3 Sec Substitution - containing x2a2Edit

 

From the diagram
 


 

 

 

 
 

 

 

 

 

 

 

 

SummaryEdit

Definition 1 Sine Definition 2 Tan Definition 3 Sec
     

This table summarises the definitions that we identify in special integral cases and how they relate to trig identities.

TechniqueEdit

Integration 1 Sine Substitution - containing a2x2Edit

We begin with the integral

 

Step 1 - Identify Trigonometric Substitution Type
We identify this integral as a trigonometric sine substitution.

Step 2 - Identifying Identities for Substitution

 
     
     
   
   

or
 

Step 3 - Substituting Identities into Integral
Now we solve the integral using the following steps

 
 
 
 
 
 
 

Step 5 - Final Substitution of  

 
 
 

Example 1 - Sec substitutionEdit

Evaluate

 


Solution

 

Step 1 - Identify Trigonometric Substitution Type

 

Step 2 - Identifying Identities for Substitution

 
         
         
         
 
     

or
 

 
 
 

 
 
 
 
 

 
 
 
 

 
 
 

 
 
 

 

 

 


 

 

 


Step 3 - Substituting Identities into Integral

 

 

 

 

 

 

 

 

Step 5 - Final Substitution of  

 

 

 

The Definite IntegralEdit