1

AntiderivativeAn antiderivative of a function f is a function F such that

F f

Ex.2( ) 3 2F x x

An antiderivative of ( ) 6f x x

since ( ) ( ).F x f x is

Lecture 17 The Indefinite IntegralWaner Section 6.1 Page 352~363

2

( )f x dx

Which means: find the set of all antiderivatives of f.

When you see this:

Think this: “the indefinite integral of f with respect to x,”

( )f x dx

Integral sign Integrand

Indefinite Integral

x is called the variable of integration

3

Every antiderivative of a function must be of the form F(x) = G(x) + C, where C is a constant.

Notice 26 3xdx x C

Constant of Integration

Represents every possible antiderivative of 6x.

4

Power Rule for the Indefinite Integral

1

if 11

nn xx dx C n

n

Ex.4

3

4

xx dx C

5

Indefinite Integral of x-1

1 1lnx dx dx x C

x

x xe dx e C Indefinite Integral of ex and bx

ln

xx bb dx C

b

6

Sum and Difference Rules

( ) ( )kf x dx k f x dx

f g dx fdx gdx Ex.

( constant)k

4 43 32 2 2

4 2

x xx dx x dx C C

2 2x x dx x dx xdx 3 2

3 2

x xC

Constant Multiple Rule

Ex.

7

Indefinite Integral Practice

dxx5 Cx

6

6

dxx 5 Cx

4

4

8

Indefinite Integral Practice

dx6 Cx 6

dxxx )( 2 xdxdxx2

Cxx

23

23

Cx

Cx

23

23

9

Indefinite Integral Practice

dxxx )( 3.13.2

Cxx

3.03.3

3.03.3

Cxx

3.03.3

3.03.3

10

Indefinite Integral Practice

dx

xxx 32

121 dxxxx 321 2

Cxx

x 22

12ln

Cxxx 21

2

1

1

2ln

11

Indefinite Integral Practice

dx

x

x3

2

dx

xx

x33

2

dxxx 32 2

dx

xx 32

21

Cxx

2

2

1

21

Cxx

2

11

12

Integral Example (Different Variable)

Ex. Find the indefinite integral:

273 2 6ue u du

u

213 7 2 6ue du du u du du

u

Cuuueu 63

273 3ln

13

Finding Cost from Marginal CostWaner pg. 359

The marginal cost to manufacture baseball caps at a production level of x caps is 3.20 – 0.001x dollars per cap, and the cost of producing 50 caps is $200. Find the cost function.

14

Position, Velocity, and Acceleration Derivative Form

If s = s(t) is the position function of an object at time t, then

Velocity = v = Acceleration = a = ds

dtdv

dt

Integral Form

( ) ( )s t v t dt ( ) ( )v t a t dt

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Position, Velocity, and AccelerationWaner pg. 360

You toss a stone upward at a speed of 30 feet per second.

a. Find the stone’s velocity as a function of time. How fast and in what direction is it going after 5 seconds?

b. Find the position of the stone as a function of time. Where will it be after 5 seconds?

c. When and where will the stone reach its zenith, its highest point?