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Section 5.3A:   Integration and the Indefinite Integral




                      Derivative              Integral




                                                          1
2
3
(Variable of integration: "   with respect to x   ".)




                                                        4
Differentiate (         verb )                                                               Integrate (     verb )

               ( )                                                                                   ( )
To find the instantaneous                                                          To find the family of functions
        rate of change                                                                       whose derivative
           of function                                                                            is function




                   Differentiation and Integration are                        ____________________                    operations.


                                                    "   undo "            each other.




                                                                                                                                    5
III. Mechanics.       The rules of integration mirror the rules of differentiation.

Differentiation                                                                        Integration

1.  Power Rule                                                                         1.  Power Rule




                                                                                                        6
Differentiation        Integration
2.  Constant Rule    2.  Constant Rule 
                                                      0
                     (Power Rule Special case for x       )




                                                              7
Differentiation                  Integration
3.  Constant times a function   3.  Constant times a function




                                                                8
Differentiation       Integration
4.  Sum of Functions   4.  Sum of Functions




                                              9
Differentiation                   Integration
5.  Natural Exponential Function   5.  Natural Exponential Function




                                                                      10
Differentiation                    Integration
6.  Base­ b Exponential Functions   6.  Base­ b Exponential Functions




                                                                        11
Differentiation          Integration

7.  Composite Functions   7.  Composite Functions
    (the Chain Rule)          ("u" Substitution)




                                                    12
13
14
15
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