Integration Using Trigonometric Substitution

1. Integration Using Trigonometric Substitution Brought to you by Tutorial Services – The Math Center

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 0 = sin 0 = • 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