1. Integration Using Trigonometric Substitution
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2. Objective
To eliminate radicals in the integrand using Trigonometric Substitution
For integrals involving use u = a sin
For integrals involving use u = a tan
For integrals involving use u = a sec

3. For integrals involving
Let u = a sin
Inside the radical you will have
Using the Pythagorean Identities, that is equal to
This will result in = a cos

4. For integrals involving
Let u = a tan
Inside the radical you will have
Using the Pythagorean Identities, that is equal to
This will result in = a sec

5. For integrals involving
Let u = a sec
Inside the radical you will have
Using the Pythagorean Identities, that is equal to
This will result in = + a tan
Positive if u > a, Negative if u < - a

6. Converting Limits
By converting limits, you avoid changing back to x, after you are done with the integration
Because has the form
then u = x, a = 3, and x = 3 sin

7. Converting Limits Now, when x = 0, the Lower Limits is: 0 = 3 sin
Now, when x = 3, the Upper Limit is:
3 = 3 sin
1 = sin
/2 =

8. Examples
Solve the following integrals:

9. Integration Using Trigonometric Substitution Links
Integration Using Trigonometric Substitution Handout
Trigonometric Identities Handout
Integrals and Derivatives Handout
Trigonometric Substitution Quiz