6.1 Antiderivatives

First, a little review:

Consider: 23y x= +

then: 2y x! = 2y x! =

25y x= !

or

It doesn’t matter whether the constant was 3 or -5, sincewhen we take the derivative the constant disappears.

!

First, a little review:

Consider: 23y x= +

then: 2y x! = 2y x! =

25y x= !

or

It doesn’t matter whether the constant was 3 or -5, sincewhen we take the derivative the constant disappears.

However, when we try to reverse the operation:

Given: 2y x! = find y

!

First, a little review:

Consider: 23y x= +

then: 2y x! = 2y x! =

25y x= !

or

It doesn’t matter whether the constant was 3 or -5, sincewhen we take the derivative the constant disappears.

However, when we try to reverse the operation:

Given: 2y x! = find y

2y x C= +

We don’t know what theconstant is, so we put “C” inthe answer to remind us thatthere might have been aconstant.

!

If we have some more information we can find C.

Given: and when , find the equation for .2y x! = y4y = 1x =

!

If we have some more information we can find C.

Given: and when , find the equation for .2y x! = y4y = 1x =

2y x C= +

!

If we have some more information we can find C.

Given: and when , find the equation for .2y x! = y4y = 1x =

2y x C= +

24 1 C= +

3 C=

23y x= +

!

If we have some more information we can find C.

Given: and when , find the equation for .2y x! = y4y = 1x =

2y x C= +

24 1 C= +

3 C=

23y x= +

This is called an initial valueproblem. We need the initialvalues to find the constant.

!

If we have some more information we can find C.

Given: and when , find the equation for .2y x! = y4y = 1x =

2y x C= +

24 1 C= +

3 C=

23y x= +

This is called an initial valueproblem. We need the initialvalues to find the constant.

An equation containing a derivative is called a differentialequation. It becomes an initial value problem when youare given the initial condition and asked to find the originalequation.

!

Integrals such as are called definite integrals

because we can find a definite value for the answer.

42

1

x dx!

42

1

x dx!

Integrals such as are called definite integrals

because we can find a definite value for the answer.

42

1

x dx!

42

1

x dx!4

3

1

1

3x C+

3 31 14 1

3 3C C

! " ! "# $ #% & % &

' '+ +

( (

64 1

3 3C C!+ !

63

3= 21=

The constant always cancelswhen finding a definiteintegral, so we leave it out!

Integrals such as are called indefinite integrals

because we can not find a definite value for the answer.

2x dx!

2x dx!

31

3x C+

When finding indefiniteintegrals, we alwaysinclude the “plus C”.

!

Many of the integral formulas are listed on page 332. Thefirst ones that we will be using are just the derivativeformulas in reverse.

CW P. 337 (2, 6, 8, 10)