8.3
Trigonometric Substitutions
Spring 2023
Haitham S. Solh
Trigonometric Substitutions
 Trigonometric substitutions occur when we replace the
variable of integration by a trigonometric functions.
 The most common substitutions are:
x = a tan θ ; x = a sin θ; and x = a sec θ
 Why and where do we substitute?
Trigonometric Substitutions
Trigonometric Substitutions
 Note: We want any substitutions to be reversible so that
we can change to the original variable afterward.
 For instance, if we set x = a tan θ, we should be able to
go back to x by setting θ = tan– 1 (x/a) after the
integration takes place. Similarly for x = a sin θ and
x = a sec θ
 What are the conditions we need to set in each case?
Procedure
Example 1
Example 1 (Continued)
Example 2
Example 3
Example 3