u-du: Integrating Composite Functions AP Calculus

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”

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

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

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

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

Example 1 : du given Ex 1:

Example 2: du given Ex 2:  

Example 3: du given Ex 3:  

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

Example 5: Regular Method  

Working with Constants < multiplying by one> Constant Property of Integration   ILL. let u = du = and becomes = Or alternately = =

Example 6 : Introduce a Constant - my method

Example 7 : Introduce a Constant

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

Example 9 : Introduce a Constant - extra constant

Example 10 : Polynomial

Example 11: Separate the numerator

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

Formal Change of Variables << the Extra “x”>> Rewrite in terms of u - du

Formal Change of Variables << the Extra “x”>> Solve for x in terms of u - du   <<alt. Method>> - could divide or multiply by

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.

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 /