1. Trigonometric Substitution
Lesson 9.6

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 a x

3. Use identity tan2x + 1 = sec2x
Example 3 x θ
Given
Consider the labeled triangle
Let x = 3 tan θ (Why?)
And dx = 3 sec2 θ dθ
Then we have
Use identity tan2x + 1 = sec2x

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

5. Knowing Which Substitution

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

7. Keep Going!
Now finish the integration

8. Application
Find the arc length of the portion of the parabola y = 10x – x2 that is above the x-axis
Recall the arc length formula

9. Assignment Lesson 9.6 Page 386
Exercises 1 – 33 (every other odd) Also 37, 39, and 41