4.1 Antiderivatives & Indefinite Integration

1. 4.1 Antiderivatives & Indefinite Integration

2. Definition of Antiderivative • A function is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I • Note: F is called an antiderivative of f, rather than the antiderivative of f because: • F1(x)=x3, F2(x)=x3 – 5 and F3(x)=x3+97 are all antiderivatives of f(x)=3x2

3. Representation of Antiderivatives • If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form: • G(x) = F(x) + c, for all x in I where c is a constant

4. Example 1:Solving a differential • Find the general solutions of the differential equation y’=2

5. Notation for antiderivatives • When solving a differential equation of the form: • It is convenient to write it in the equivalent form: • dy = f(x) dx • The process of finding all solutions of this equation is called antidifferentiation (or indefinite integration)

6. Notation for Antiderivatives • Antidifferentiation is denoted by the integral sign: • The general solution is denoted by: Variable of Integration Constant of Integration Integrand

7. Basic Integration Rules Moreover. If , then

8. Basic Integration Rules • Look at rules on page 244

9. Power Rule

10. Example 1

11. Example 2

12. Example 3

13. Example 4

14. Example 5

15. Example 6

16. Example 7

17. Example 8