Understanding Antiderivatives in Calculus

1. Section 17.4IntegrationLAST ONE!!!Yah Buddy!

2. Introduction • A physicist who knows the velocity of a particle might wish to know its position at a given time. • A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time.

3. Antiderivatives • In each case, the problem is to find a function F whose derivative is a known function f. • If such a function F exists, it is called an antiderivativeof f. Definition A function F is called an antiderivative of f on an interval I if F’(x) = f (x) for all x in I.

4. Find the integral. (Find the antiderivative.) = ?

5. Antiderivatives • If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + C where C is an arbitrary constant. Theorem • Going back to the function f (x) = x2, we see that the general antiderivative of f is ⅓ x3 + C.

6. Notation for Antiderivatives • The symbol is traditionally used to represent the most general an antiderivative of f on an open interval and is called the indefinite integral of f . • Thus, means F’(x) = f (x) is because the derivative of

7. Constant of Integration Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Example: Represents every possible antiderivative of 6x.

8. Power Rule for the Indefinite Integral Example:

9. Power Rule for the Indefinite Integral Indefinite Integral of ex and bx

10. Sum and Difference Rules Example:

11. Constant Multiple Rule Example:

12. Integration by Substitution Method of integration related to chain rule. If u is a function of x, then we can use the formula

13. Integration by Substitution Example:Consider the integral: Sub to get Integrate Back Substitute

14. Example:Evaluate Pick u, compute du Sub in Integrate Sub in

15. Example:Evaluate

16. Examples on your own:

17. Find the integral of each: 1.) 2.) 3.) 4.)

18. Find the integral of each: 5.) 6.) 7.) 8.)

19. Find the integral of each: 9.) 10.) 11.) 12.)

20. Find the integral of each: 13.) 14.)