Section 15.1 - Antiderivatives

Antiderivatives

Definition A function F(x) is called an antiderivative of f(x) if F ′(x) = f (x).

Examples: What’s the antiderivative of f(x) = 1/x ?What’s the antiderivative of g(x) = 2x ?What’s the antiderivative of h(x) = 4 ?

Antiderivatives

Theorem If F(x) is an antiderivative of f (x), then the most general antiderivative of f(x) is the family of functions given by F(x) + c, where c is an arbitrary constant.

Example: Determine the most general antiderivative of the following functions.

23)(

)(

xxg

exf x

Notations for AntiderivativesDefinition The operation of finding the antiderivatives of a function f(x) is called antidifferentiation (or Integration) and is denoted by the integral sign

dxxf )( CxF )(

Variable of integration

Constant of integrationintegrandIntegral sign

Antiderivative

Examples

3

4

2

4

dx

x dx

x dx

Integration Formula

dxx

dxk

dxx

dxe

n

x

1

Cn

x

Ckx

Cx

Ce

n

x

1

ln

1

Integration Formula

( ) ( )

[ ( ) ( )] ( ) ( )

k f x dx k f x dx

f x g x dx f x dx g x dx

Examples

344

4 3

4

7 5

47

1

2 5 3

2

xe dx

t dtt

x dxx

t tdt

t

Examples

1) Determine f (x) if 1)0(6)(' 2 fandxxf

2) Find the cost function for each marginal cost function.

C’(x) = 5x – 1/x ; 10 units cost $94.20