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Trigonometric Substitution

In Lesson 9.6, we delve into a new set of patterns for trigonometric substitution within integrals, emphasizing the importance of right triangles. By drawing and labeling triangles that reveal relationships between variables like (a), (x), and (theta), we can better understand integration. Utilizing the identity (tan^2 x + 1 = sec^2 x), we explore substitutions like (x = 3tan theta) and distinguish among different triangles to simplify integration tasks. Application involves finding the arc length of certain parabolic segments, enhancing our comprehension of these mathematical concepts.

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Trigonometric Substitution

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  1. Trigonometric Substitution Lesson 9.6

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

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