Trig%20Substitution

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Lesson 7-3

• Trig Substitution

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Table of Trigonometric Substitutions
Expression Substitution Trig Identity
a² - x² x a sin ? -p/2 ? p/2 1 - sin² ? cos² ?
a² x² x a tan ? -p/2 ? p/2 1 tan² ? sec² ?
x² - a² x a sec ? 0 ? p/2 or p ? 3p/2 sec²? 1 tan² ?
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Type 1 ?a²- x²

• Sub x a sin ? and dx a cos ? d?
• Square root reduces to a cos ?
• Integrate
• Sub back in x

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7-3 Example 1
? ?4 - x² dx
x 2 sin ? dx 2 cos ? d? Use Trig id sin²
? 1 - cos² ?
? 2(cos ?) (2cos ? d?) 4 ? cos² ? d?
4( ½ ? ¼ sin2? ) C 2? 2sin ? cos ? C
Double Angle formula
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7-3 Example 2
x2 dx ----------- (4- x2)3/2
?
Let x 2 sin ? dx 2 cos ? d?
Trig Reduction Formula
tan ? - ? C
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Type 2 ?a² x²

• Sub x a tan ? and dx a sec² ? d?
• Square root reduces to a sec ?
• Integrate trig function
• Sub back in x

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7-3 Example 3
dx ----------- x2 9
?
Let x 3 tan ? dx 3 sec² ? d?
(1/3) ? C
1 x ---- tan-1 --- C
3 3
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7-3 Example 4
? ?4 x² dx
? ?4 4tan² x (2 sec2 ?) d?
Let x 2 tan ? and dx 2 sec2 ? d?
2 ? sec ? sec² ? d?
Using Trig Reduction Formula
From Table of Integrals
½ sec x tan x ½ ln sec x tan x C
½(?4x² /2)(x/2) ½ ln (?4x² /2)
(x/2) C
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Type 3 ? x² - a²

• Sub x a sec ? and dx a sec ? tan ? d?
• Square root reduces to a tan ?
• Integrate trig function
• Sub back in x

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7-3 Example 5
dx ---------------- x2 (x2
9)½
?
let x 3 sec ? and dx 3 sec ? tan ? d?
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7-3 Example 6
(x2 16)½ ----------- dx x
?
let x 4 sec ? and dx 4 sec ? tan ? d?
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Summary Homework

• Summary
• Trig Substitution can allow us to solve some hard
  integrals involving square roots
• Basic steps the same (but different
  substitutions)
• Substitute to eliminate square root
• Evaluate the trigonometric integral
• Convert back to original variable using triangle
• All based on Geometric Right Triangle Trig Dfns
• Homework
• pg 488-489, Day 1 1, 2, 5, 9, 10 Day 2 3,
  7, 11, 14, 17