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# Trigonometric Substitution - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Trigonometric Substitution' - rea

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### Trigonometric Substitution

Lesson 8.4

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

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

3

x

Finishing Up

• Our results are in terms of θ

• We must un-substitute back into x

• Use the triangle relationships

• For each problem, identify which substitution and which triangle should be used

• Now finish the integration

• Find the arc length of the portion of the parabola y = 10x – x2 that is above the x-axis

• Recall the arc length formula

• Useful formulas from Theorem 8.2

• Look for these patterns and plug in thea2 and u2 found in your particular integral

• Lesson 8.4

• Page 550

• Exercises 1 – 45 EOOAlso 67, 69, 73, and 77