1. Warm Up
Before you start pg 342

2. 7.1 Antiderivatives and Indefinite Integration
IB Math E09

3. Definition of an Antiderivative
A function F is an antiderivative of f on an interval I if F’(x)=f(x) for all x in I.
If f(x)=12x, F(x)=6x2+9, F(x)=6x2-2
and F(x)=6x2 are all antiderivatives of f(x)

4. Theorem 4.1 Representation of Antiderivatives
Recall:
If f(x)=12x, F(x)=6x2+9, F(x)=6x2-2
and F(x)=6x2 are all antiderivatives of f(x)
Theorem: 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.
C is called the “constant of integration”

5. Example 1. Find the general solution of the differential equation
A) y’=8
B) y’=2x
C) y’=3x-4

6. Notation Recall the derivative is often represented as
For integration, it is convenient to writ the equialent differential form: dy=f(x)dx
The operation of finding all solutions of this equation (dy=f(x)dx) is called antidifferentiation or indefinite integration and is denoted:
Example:

7. One basic integration rule is:
Can you generate a list of other integration rules based on your current knowledge of differentiation rules?

8. Basic Integration Rules
You can separate added terms in an integral.

9. Example: Find the antiderivative of:

10. Example: Rewriting Before Integrating

11. Ex: Integrating a polynomial function

12. Ex: Rewriting before Integrating

13. Ex: Rewriting Before Integrating

14. Method of Substitution

15. Ex:Initial Conditions and Particular Solutions
Find the general solution of: F’(s)=4s-9s2
And find the particular solution that satisfies the initial condition F(3)=1.

16. Method of Substitution
Investigation page 348 “Integration (ax+b)n”

18. Homework
7A (1-8)
7B (1-8)
7C (1-6)
7D (1-7)