Antiderivatives and Indefinite Integrals

1. Antiderivatives and Indefinite Integrals Modified by Mrs. King from Paul's Online Math Tutorials and Notes http://tutorial.math.lamar.edu/AllBrowsers/2413/ IndefiniteIntegrals.asp

2. Introduction • In the past two chapters we’ve been given a function f(x) and asking what the derivative of this function was.  • We now want to turn things around and ask what function we differentiated to get the function f(x).

3. Example 1 What function did we differentiate to get the following function:

4. Process for finding an Antiderivative: • This is the reverse of differentiation, so we are going to add oneto the exponent and then divide by that new exponent.

5. We found the correct function.Or did we? • We know that the derivative of a constant is zero and any function of the form will result in the function f(x) upon differentiating.

6. Definitions • Given a function f(x) an anti-derivative of f(x) is any function F(x) such that

7. Definitions • If F(x) is any anti-derivative of f(x) then the most general anti-derivative of f(x) is called an indefinite integral and denoted

8. Definitions • In this definition the ∫ is called the integral symbol, f(x) is called the integrand, x is called the integration variable and the “c” is called the constant of integration.

9. Example #2

10. Example #3

11. Example #4

12. Example #5

13. Example #6

14. Homework Page 255 #9, 10, 12, 16-28 even (no check), 35, 36