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**u-du: Integrating Composite Functions**

AP Calculus

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**Integrating Composite Functions**

(Chain Rule) Remember: Derivatives Rules Remember: Layman’s Description of Antiderivatives *2nd meaning of “du” du is the derivative of an implicit “u”

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**Integrating Composite Functions**

u-du Substitution Integrating Composite Functions (Chain Rule) Revisit the Chain Rule If let u = inside function du = derivative of the inside becomes

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Development from the layman’s idea of antiderivative “The Family of functions that has the given derivative” must have the derivative of the inside in order to find the antiderivative of the outside

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A Visual Aid USING u-du Substitution a Visual Aid REM: u = inside function du = derivative of the inside let u = becomes now only working with f , the outside function

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**Working With Constants: Constant Property of Integration**

With u-du Substitution REM: u = inside function du = derivative of the inside Missing Constant? u = du = Worksheet - Part 1

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Example 1 : du given Ex 1:

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Example 2: du given Ex 2:

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Example 3: du given Ex 3:

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Example 4: du given Ex 4: TWO WAYS! Differ by a constant

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**Example 5: Regular Method**

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**Working with Constants < multiplying by one>**

Constant Property of Integration ILL let u = du = and becomes = Or alternately = =

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**Example 6 : Introduce a Constant - my method**

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**Example 7 : Introduce a Constant**

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**Example 8 : Introduce a Constant << triple chain>>**

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**Example 9 : Introduce a Constant - extra constant**

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Example 10 : Polynomial

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**Example 11: Separate the numerator**

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**Formal Change of Variables << the Extra “x”>> **

Solve for x in terms of u ILL: Let Solve for x in terms of u then and becomes

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**Formal Change of Variables << the Extra “x”>> **

Rewrite in terms of u - du

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**Formal Change of Variables << the Extra “x”>>**

Solve for x in terms of u - du <<alt. Method>> - could divide or multiply by

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**Complete Change of Variables << Changing du >>**

At times it is required to even change the du as the u is changed above. We will solve this later in the course.

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Development must have the derivative of the inside in order to find the antiderivative of the outside *2nd meaning of “dx” dx is the derivative of an implicit “x” more later if x = f then dx = f /

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