Warm Up Before you start pg 342

7.1 Antiderivatives and Indefinite Integration IB Math E09

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)

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”

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

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:

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

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

Example: Find the antiderivative of:

Example: Rewriting Before Integrating

Ex: Integrating a polynomial function

Ex: Rewriting before Integrating

Ex: Rewriting Before Integrating

Method of Substitution

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.

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

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