CHAPTER 4 SECTION 4.5 INTEGRATION BY SUBSTITUTION

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CHAPTER 4SECTION 4.5INTEGRATION BY SUBSTITUTION
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Theorem 4.12 Antidifferentiation of a Composite
Function
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Substitution with Indefinite Integration

• This is the backwards version of the chain rule
  
• Recall
• Then

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Substitution with Indefinite Integration

• In general we look at the f(x) and split it
• into a g(u) and a du/dx
• So that

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Substitution with Indefinite Integration

• Note the parts of the integral from our example

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Substitution with Indefinite Integration

• Let u So, du (2x -4)dx

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Guidelines for Making a Change of Variables
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Theorem 4.13 The General Power Rule for
Integration
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Example 1
The variable of integration must match the
variable in the expression.
Dont forget to substitute the value for u back
into the problem!
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Example 2
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Example 3
Solve for dx.
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Example 4
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Example 5
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Example 6
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Can You Tell?

• Which one needs substitution for integration?

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Integration by Substitution
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Integration by Substitution
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Solve the differential equation
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Solve the differential equation
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Theorem 4.14 Change of Variables for Definite
Integrals
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or you could convert the bound to us.
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Example 7
We can find new limits, and then we dont have to
substitute back.
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Example 9
Dont forget to use the new limits.
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Theorem 4.15 Integration of Even and Odd Functions
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Even/Odd Functions
If f(x) is an even function, then
If f(x) is an odd function, then
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Even/Odd Functions
If f(x) is an even function, then
If f(x) is an odd function, then
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