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Trigonometric Substitution

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

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  1. Trigonometric Substitution Lesson 8.4

  2. 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

  3. 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

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

  5. Knowing Which Substitution u u

  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. Special Integration Formulas • Useful formulas from Theorem 8.2 • Look for these patterns and plug in thea2 and u2 found in your particular integral

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

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