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Trigonometric Substitution. Lesson 8.4. a. x. 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. θ. 3. x.

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Presentation Transcript
new patterns for the integrand

a

x

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
example

θ

3

x

Use identitytan2x + 1 = sec2x

Example
  • Given
  • Consider the labeled triangle
    • Let x = 3 tan θ (Why?)
    • And dx = 3 sec2θ dθ
  • Then we have
finishing up

θ

3

x

Finishing Up
  • Our results are in terms of θ
    • We must un-substitute back into x
    • Use the triangle relationships
try it
Try It!!
  • For each problem, identify which substitution and which triangle should be used
keep going
Keep Going!
  • Now finish the integration
application
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
special integration formulas
Special Integration Formulas
  • Useful formulas from Theorem 8.2
  • Look for these patterns and plug in thea2 and u2 found in your particular integral
assignment
Assignment
  • Lesson 8.4
  • Page 550
  • Exercises 1 – 45 EOOAlso 67, 69, 73, and 77
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