Antiderivative. Buttons on your calculator have a second button Square root of 100 is 10 because...

AntiderivativeAntiderivative

Buttons on your Buttons on your calculator have a second calculator have a second buttonbutton

Square root of 100 is 10 because Square root of 100 is 10 because 10 square is 10010 square is 100 Arcsin( ½ ) = Arcsin( ½ ) = /6 because/6 because Sin( Sin( /6 ) = ½ /6 ) = ½ Reciprocal of 20 is 0.05 becauseReciprocal of 20 is 0.05 because Reciprocal of 0.05 is 20Reciprocal of 0.05 is 20

Definition:Definition:

F is an antiderivative of f on D if f is F is an antiderivative of f on D if f is the derivative of F on D or F ' = f the derivative of F on D or F ' = f on D.on D.

Definition:Definition:

F is an antiderivative of f on D if f is F is an antiderivative of f on D if f is the derivative of F on D or F ' = f the derivative of F on D or F ' = f on D.on D.

Instead of askingInstead of asking What is the derivative of F? f What is the derivative of F? f We askWe ask What is the antiderivative of f? F What is the antiderivative of f? F

Definition:Definition:

Name an antiderivative of f(x) = Name an antiderivative of f(x) = cos(x).cos(x).

F(x) = sin(x) because the derivative F(x) = sin(x) because the derivative ofof

sin(x) is cos(x) or F’(x) = f(x).sin(x) is cos(x) or F’(x) = f(x).

G = F + 3 is also an G = F + 3 is also an antiderivative of fantiderivative of f because G' = (F + 3)' = f + 0 = f.because G' = (F + 3)' = f + 0 = f. sin(x) and the sin(x) + 3 are both sin(x) and the sin(x) + 3 are both

antiderivatives of the Cos(x) on R antiderivatives of the Cos(x) on R because when you differentiate because when you differentiate either one you get cos(x). either one you get cos(x).

G = F + k is also an G = F + k is also an antiderivative of fantiderivative of f

How many antiderivatives of How many antiderivatives of Cos(x) are there? Cos(x) are there?

If you said infinite, you are If you said infinite, you are correct. correct.

G(x) = sin(x) + k is one for every G(x) = sin(x) + k is one for every real number k. real number k.

Which of the following is Which of the following is an antiderivative of y = an antiderivative of y = cos(x)?cos(x)?

A.A. F(x) = 1 – sin(x)F(x) = 1 – sin(x)

B.B. F(x) = - sin(x)F(x) = - sin(x)

C.C. F(x) = sin(x) + 73F(x) = sin(x) + 73

D.D. F(x) = cos(x)F(x) = cos(x)

Which of the following is Which of the following is an antiderivative of y = an antiderivative of y = cos(x)?cos(x)?

A.A. F(x) = 1 – sin(x)F(x) = 1 – sin(x)

B.B. F(x) = - sin(x)F(x) = - sin(x)

C.C. F(x) = sin(x) + 73F(x) = sin(x) + 73

D.D. F(x) = cos(x)F(x) = cos(x)

Buttons on your Buttons on your calculator have a second calculator have a second buttonbutton

A square root of 100 is -10 A square root of 100 is -10 because because

(-10) square is 100(-10) square is 100 arcsin( ½ ) = 5arcsin( ½ ) = 5/6 because/6 because Sin( 5Sin( 5/6 ) = ½ /6 ) = ½ Recipicol of 20 is 0.05 becauseRecipicol of 20 is 0.05 because Recipicol of 0.05 is 20Recipicol of 0.05 is 20

Theorem:Theorem:

If F and G are antiderivatives of f If F and G are antiderivatives of f on an interval I, then F(x) - G(x) = on an interval I, then F(x) - G(x) = k where k is a real number.k where k is a real number.

Proof: Proof:

Let F and G be antiderivatives of f Let F and G be antiderivatives of f on I and define H(x) = F(x) - G(x) on I and define H(x) = F(x) - G(x) on I.on I.

Proof: Proof:

Let F and G be antiderivatives of f Let F and G be antiderivatives of f on I and define H(x) = F(x) - G(x) on I and define H(x) = F(x) - G(x) on I.on I.

H’(x) = F’(x) – G’(x) = f(x) – f(x) = H’(x) = F’(x) – G’(x) = f(x) – f(x) = 0 for every x in I. 0 for every x in I.

What if H(d) is different than What if H(d) is different than H(e)?H(e)?

H(x) = F(x) - G(x) = k H(x) = F(x) - G(x) = k

If the conclusion of the theorem If the conclusion of the theorem were false, there would be were false, there would be numbers d < e in I for which H(d) numbers d < e in I for which H(d) H(e). H(e).

H(x) = F(x) - G(x) = k H(x) = F(x) - G(x) = k

Since H is differentiable and Since H is differentiable and continuous on [d, e], the Mean continuous on [d, e], the Mean Value Theorem guarantees a c, Value Theorem guarantees a c, between d and e, for which H'(c) between d and e, for which H'(c) = =

(H(e) - H(d))/(e - d) which can’t be (H(e) - H(d))/(e - d) which can’t be 0.0.

H(x) = F(x) - G(x) = kH(x) = F(x) - G(x) = kH’(x) = 0 for all x in I H’(x) = 0 for all x in I

H’(c) = (H(e) - H(d))/(e - d)H’(c) = (H(e) - H(d))/(e - d)

This contradicts the fact that H’( c) This contradicts the fact that H’( c) must be zero.must be zero.

q.e.d.q.e.d.

Which of the following Which of the following are antiderivatives of y = are antiderivatives of y = 4?4?A.A. 4x4x

B.B. 4x + 24x + 2

C.C. 4x - 74x - 7

D.D. All of the aboveAll of the above

Which of the following Which of the following are antiderivatives of y = are antiderivatives of y = 4?4?A.A. 4x4x

B.B. 4x + 24x + 2

C.C. 4x - 74x - 7

D.D. All of the aboveAll of the above

Since antiderivatives differ by a Since antiderivatives differ by a constant on intervals, we will use constant on intervals, we will use the notation  f(x)dx to represent the notation  f(x)dx to represent the family of all antiderivatives of the family of all antiderivatives of f. When written this way, we call f. When written this way, we call this family the indefinite integral this family the indefinite integral of f. of f.

( )f x dx

Using our new notation, evaluate Using our new notation, evaluate

( )f x dx

cos( )x dx sin( )x C

Using our new notation, evaluate Using our new notation, evaluate

( )f x dx

4dx 4x C

==

A.A. TrueTrue

B.B. FalseFalse

sin(2 )x dx2[sin( )]x c

==

A.A. TrueTrue

B.B. FalseFalse

sin(2 )x dx2[sin( )]x c

Theorems Theorems

because [x + c]’ = 1because [x + c]’ = 1

What is the integral of dx?What is the integral of dx?

x grandpa, xx grandpa, x

1dx x c

dx x c

..

A.A. cc

B.B. 3x + c3x + c

C.C. xx2 2 + c+ c

D.D. x + cx + c

dx

..

A.A. cc

B.B. 3x + c3x + c

C.C. xx2 2 + c+ c

D.D. x + cx + c

dx

Theorems Theorems

If F is an antiderivative of f => F’=fIf F is an antiderivative of f => F’=f

andand

kF is an antiderivative of kfkF is an antiderivative of kf

because [kF]’ = k[F]’ = k fbecause [kF]’ = k[F]’ = k f

andand

( ) ( )f x dx F x c

( ) ( )kf x dx k f x dx

( ) ( )kf x dx kF x c

( ) [ ( ) ]k f x dx k F x c

Theorems Theorems

3cos( ) 3 cos( )x dx x dx

( ) ( )kf x dx k f x dx

edx e dx e

3sin( )x c

x c

..

A.A. 12 x + c12 x + c

B.B. 00

C.C. - 24 x + c- 24 x + c

D.D. 24 x + c24 x + c

24dx

..

A.A. 12 x + c12 x + c

B.B. 00

C.C. - 24 x + c- 24 x + c

D.D. 24 x + c24 x + c

24dx

Theorem Theorem

Proof: Proof: F(x)F(x)

F’(x) = F’(x) =

1

1

nn xx dx c

n

1

1

nxc

n

( 1)1

nxn

n

nx

Examples Examples

1

1

nn xx dx c

n

4x dx 4 1

4 1

xc

5

5

xc

47x dx 5

75

xc

..

A.A. 3 x3 x22 + c + c

B.B. 9 x9 x22 + c + c

C.C. xx33 + c + c

D.D. 3 x3 x33 + c + c

23x dx

..

A.A. 3 x3 x22 + c + c

B.B. 9 x9 x22 + c + c

C.C. xx33 + c + c

D.D. 3 x3 x33 + c + c

23x dx

Examples Examples

1

1

nn xx dx c

n

44

1dx x dx

x

4 1

4 1

xc

3

3

xc

31 227 7 7

32

xxdx x dx c

314

3

xc

3

1

3c

x

..

A.A. 4/x + c4/x + c

B.B. -4/x + c-4/x + c

C.C. 4/x4/x33 + c + c

D.D. -4/x-4/x33 + c + c

2

4dx

x

..

A.A. 4/x + c4/x + c

B.B. -4/x + c-4/x + c

C.C. 4/x4/x33 + c + c

D.D. -4/x-4/x33 + c + c

2

4dx

x

Theorem Theorem

If h(x) = 2cos(x) + 3xIf h(x) = 2cos(x) + 3x22 – 4, – 4,

evaluate evaluate

( ) ( ) ( ) ( )f x g x dx f x dx g x dx

( )h x dx 22cos( ) 3 4x x dx

22 cos( ) 3 4x dx x dx dx 2sin( ) 3x

22cos( ) 3 4x x dx

3

43

x x c

1 12 52 22 sec ( ) 3x dx x dx x dx x dx

2 tan( ) 3x

25

1 12sec ( ) 3x x dx

x x

3

2

32

x

4

4

x

1

2

12

xc

3

24

12 tan( ) 2

4x x

x 2 x c

Trigonometry Trigonometry TheoremsTheoremssin( ) cos( )x dx x c cos( ) sin( )x dx x c

2sec ( ) tan( )x dx x c 2csc ( ) cot( )x dx x c

sec( ) tan( ) sec( )x x dx x c csc( )cot( ) csc( )x x dx x c

..

A.A. 2 sin(x) – 1/x + c2 sin(x) – 1/x + c

B.B. 2 sin(x) – 1/x2 sin(x) – 1/x33 + c + c

C.C. - 2 sin(x) – 1/x + c- 2 sin(x) – 1/x + c

2

12cos( )x dx

x

..

A.A. 2 sin(x) – 1/x + c2 sin(x) – 1/x + c

B.B. 2 sin(x) – 1/x2 sin(x) – 1/x33 + c + c

C.C. - 2 sin(x) – 1/x + c- 2 sin(x) – 1/x + c

2

12cos( )x dx

x

Trigonometry Trigonometry TheoremsTheorems

1sin( ) cos( )kx dx kx c

k 1

cos( ) sin( )kx dx kx ck

2 1

sec ( ) tan( )kx dx kx ck

2 1

csc ( ) cot( )kx dx kx ck

1sec( ) tan( ) sec( )kx kx dx kx c

k 1

csc( )cot( ) csc( )kx kx dx kx ck

..

A.A. sin(2x) – 1/x + csin(2x) – 1/x + c

B.B. 4 sin(2x) – 1/x + c4 sin(2x) – 1/x + c

C.C. -2 sin(2x) – 1/x + c-2 sin(2x) – 1/x + c

2

12cos(2 )x dx

x

..

A.A. sin(2x) – 1/x + csin(2x) – 1/x + c

B.B. 4 sin(2x) – 1/x + c4 sin(2x) – 1/x + c

C.C. -2 sin(2x) – 1/x + c-2 sin(2x) – 1/x + c

2

12cos(2 )x dx

x