1. Trigonometric Substitution Lesson 8.4

2. New Patterns for the Integrand Now we will look for a different set of patterns And we will use them in the context of a right triangle Draw and label the other two triangles which show the relationships of a and x 2 a x

3. Example Given Consider the labeled triangle  Let x = 3 tan θ(Why?)  And dx = 3 sec 2 θ dθ Then we have 3 3 x θ Use identity tan 2 x + 1 = sec 2 x

4. Finishing Up Our results are in terms of θ  We must un-substitute back into x  Use the triangle relationships 4 3 x θ

5. Knowing Which Substitution 5 u u

6. Try It!! For each problem, identify which substitution and which triangle should be used 6

7. Keep Going! Now finish the integration 7

8. Application Find the arc length of the portion of the parabola y = 10x – x 2 that is above the x-axis Recall the arc length formula 8

9. Special Integration Formulas Useful formulas from Theorem 8.2 Look for these patterns and plug in the a 2 and u 2 found in your particular integral

10. Assignment Lesson 8.4 Page 550 Exercises 1 – 45 EOO Also 67, 69, 73, and 77 10