Trigonometric Substitutions
Contents
Introduction to this topic[edit]
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 Knowledge[edit]
 Basic Differentiation
 Basic Integration Methods
 Pythagoras Theorem
Theory of Trigonometric Substitutions[edit]
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 Theorem[edit]
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 a^{2} − x^{2}[edit]
From the diagram 


Definition 2 Tan Substitution  containing a^{2} + x^{2}[edit]
From the diagram 

Definition 3 Sec Substitution  containing x^{2} − a^{2}[edit]
From the diagram



Summary[edit]
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.
Technique[edit]
Integration 1 Sine Substitution  containing a^{2} − x^{2}[edit]
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 substitution[edit]
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