Antiderivatives

Antiderivatives

• Mr. Baird knows the velocity of particle and wants to know its position at a given time

• Ms. Bertsos knows the rate a population of bacteria is increasing and she wants to know what the size of the population will be at a future time.

• In each case the rate of change (the derivative) is known….but what is the original function?

• The original function is called the

ANTIDERIVATIVEANTIDERIVATIVE of the rate of change.

A function is called an

antiderivative of on an

interval if

for all x in .

Ff

II F x f x

DEFINITION

Suppose

What is its antiderivative?

2f x x

We can make some guesses

2 2x 2 3x

2x

2 100000000x

They all fit!

Theorem

If is an antiderivative of on

an interval , then the most

general antiderivative

of on

is

where is an arbitrary constant.

F f

II

f

F x C

C

Finding an antiderivative is also known as Indefinite Integration and the Antiderivative is the Indefinite Integral (Especially for us old guys!)

And the symbol for integration is an elongated S

More on why it’s an S later!

f x dx F x C

Integrand

Variable of Integration

Constant of Integration

This is read: The antiderivative of f with respect to x or the indefinite integral of f with respect to x is equal to…..

What is the Antiderivative ofcos ?x

cosdy

xdx

cosdy xdx cosdy xdx

siny x C

Derivative F x

We “kinda” multiply

Take the integral ofboth sides

We know what to differentiate to get cos x

Some General Rules

They are just the derivative rules in reverse

Differentiation Formula Integration Formula

0dC

dx 0dx C

dkx k

dx k dx kx C

dk f x k f x

dx k f x dx k f x dx

“Pulling out a konstant”

Some General Rules

Differentiation Formula Integration Formula

df x g x f x g x

dx

f x g x dx f x dx g x dx

1n ndx nx

dx

1

1

nn xx dx C

n

Sum / Difference Rule for Integrals

Power Rule for Integrals

Some General Rules

Differentiation Formula Integration Formula

sin cosd

x xdx

cos sinxdx x C

cos sind

x xdx

sin cosxdx x C

All the other trig functions follow