6.1 Antiderivatives and Indefinite Integration

Objectives: 1.) Understand the concept of a antiderivative 2.) Use differentiation rules to produce and use

antidifferentiation rules.

Vocab

• Differential: the differential is an equation that relates the change in y with respect to the change in x.

dy = f’(x)dx

Vocabulary

f(x)= x3 + 2x

f’(x)= 3x2 + 2

What was the function that WAS derived to get this?

We are going to start going backwards now. We are going to UNDERIVE functions…

The antiderivative function, notated BIG F, is the fuction that was derived to get a function f.

F’(x) = f(x) for all x in I

f(x)

F(x)

Integration and antidifferentiation mean the same thing

• The process of underiving

Notation

CxFxdxfy )()(

Notation/Representation

• We call G(x) the general antiderivative of f.G(x) = F(x) + C for all x in I the indefinite integral .

• C is called the constant of integration. It is the constant number that could have been wiped out in differentiation. When we antidifferentiate, we need to consider a constant may have been there.

• Consider f(x) = x2 + 1

• General antiderivative and General Solution are synonomous.

Anyone of these graphs could have produced f’(x) = 2x

Basic Rules of Integration (pg. 390)Integration of ZERO

Integration of a constant

Integration of a power

Integration with a scalar multiple

Integration of sums and differences

Homework:

• Pg 394 #1; 4; 9-13(odd); 19-23(odd); 26; 28-31; 37-39;

42; 43

Examples

Objectives

1.) Apply integration to vertical motion functions…

2.) Start thinking forwards… backwards.

Particular Solution vs. General Solution

• http://www.mathworksheetsgo.com/tools/free-online-graphing-calculator.php