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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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Trigonometric substitution

Trigonometric Substitution

Lesson 8.4


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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