Differential Equations and Mathematical Modeling

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AP CALCULUS AB

• Chapter 6
• Differential Equations and Mathematical Modeling
• Section 6.2
• Antidifferentiation by Substitution

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What youll learn about

• Indefinite Integrals
• Leibniz Notation and Antiderivatives
• Substitution in Indefinite Integrals
• Substitution in Definite Integrals
• and why
• Antidifferentiation techniques were historically
  crucial for applying the results of calculus.

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Section 6.2 Antidifferentiation by Substitution

• Definition The set of all antiderivatives of a
  function f(x) is the indefinite integral of f
  with respect to x and is denoted by

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Section 6.2 Antidifferentiation by Substitution

• is read The indefinite integral of f with
  respect to x is F(x) C.
• Example

integrand
constant of integration
integral sign
variable of integration
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Section 6.2 Antidifferentiation by Substitution

• Integral Formulas
• Indefinite Integral Corresponding
  Derivative Formula

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Section 6.2 Antidifferentiation by Substitution

• More Integral Formulas
• Indefinite Integral Corresponding
  Derivative Formula
• 4.
• 5.

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Section 6.2 Antidifferentiation by Substitution

• More Integral Formulas
• Indefinite Integral Corresponding
  Derivative Formula
• 6.
• 7.
• 8.
• 9.

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Trigonometric Formulas

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Section 6.2 Antidifferentiation by Substitution

• Properties of Indefinite Integrals
• Let k be a real number.
• Constant multiple rule
• Sum and Difference Rule

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Section 6.2 Antidifferentiation by Substitution

• Example

Cs can be combined into one big C at the end.
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Example Evaluating an Indefinite Integral

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Section 6.2 Antidifferentiation by Substitution

• Remember the Chain Rule for Derivatives
• By reversing this derivative formula, we obtain
  the integral formula

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Section 6.2 Antidifferentiation by Substitution

• Power Rule for Integration
• If u is any differentiable function of x, then

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Exponential and Logarithmic Formulas

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Section 6.2 Antidifferentiation by Substitution

• A change of variable can often turn an unfamiliar
  integral into one that we can evaluate. The
  method for doing this is called the substitution
  method of integration.

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Section 6.2 Antidifferentiation by Substitution

• Example

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• Rules For Substitution-- A mnemonic help
• L - Logarithmic functions ln x, logb x, etc.
• I - Inverse trigonometric functions arctan x,
  arcsec x, etc.
• A - Algebraic functions x2, 3x50, etc.
• T - Trigonometric functions sin x, tan x, etc.
• E - Exponential functions ex, 19x, etc.
• The function which is to be dv is whichever comes
  last in the list functions lower on the list
  have easier antiderivatives than the functions
  above them.
• The rule is sometimes written as "DETAIL" where D
  stands for dv.

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To demonstrate the LIATE rule, consider the
integral Following the LIATE rule, u x
and dv cos x dx, hence du dx and v sin x,
which makes the integral become which equals
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Section 6.2 Antidifferentiation by Substitution

• Example

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Example Paying Attention to the Differential

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Example Using Substitution

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Example Using Substitution

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Example Setting Up a Substitution with a
Trigonometric Identity

Hint let u cos x and du sinxdx
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Section 6.2 Antidifferentiation by Substitution

• Substitution in Definite Integrals
• Substitute
• and integrate with respect to u from

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Section 6.2 Antidifferentiation by Substitution

• Ex

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Example Evaluating a Definite Integral by
Substitution

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Section 6.2 Antidifferentiation by Substitution

• You try

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Section 6.2 Antidifferentiation by Substitution

• You try

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Section 6.2 Antidifferentiation by Substitution

• You try

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Section 6.2 Antidifferentiation by Substitution

• You try