Antiderivatives An antiderivative of f(x) is any function F(x) such that F’(x) = f(x)

Antiderivatives

• An antiderivative of f(x) is any function F(x) such that F’(x) = f(x)

Example

• Find antiderivatives of f(x) = x2

Example

• Find antiderivatives of f(x) = 2x

Example

• Find antiderivatives of f(x) = 1/x

Theorem

• If F(x) is an antiderivative of f(x)

then F(x) + C is an antiderivative of f(x) for any constant C

Antiderivatives Graphically

• Match the function to its antiderivative

f(x) F(x)1)

2)

3)

4) D

C

B

A

The Fundamental Theorem of Calculus, Part 1

If f is continuous on , then the function ,a b

x

aF x f t dt

has a derivative at every point in , and ,a b

x

a

dF df t dt f x

dx dx

x

a

df t dt f x

dx

First Fundamental Theorem:

1. Derivative of an integral.

a

xdf t dt

xf x

d

2. Derivative matches upper limit of integration.

First Fundamental Theorem:

1. Derivative of an integral.

a

xdf t dt f x

dx

1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

First Fundamental Theorem:

x

a

df t dt f x

dx

1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

New variable.

First Fundamental Theorem:

cos xd

t dtdx cos x 1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

The long way:First Fundamental Theorem:

20

1

1+t

xddt

dx 2

1

1 x

1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

Example

• If

find F’(x)

2

1

2 3x

F x t t dt

2

0cos

xdt dt

dx

2 2cosd

x xdx

2cos 2x x

22 cosx x

The upper limit of integration does not match the derivative, but we could use the chain rule.

53 sin

x

dt t dt

dxThe lower limit of integration is not a constant, but the upper limit is.

53 sin xdt t dt

dx

3 sinx x

We can change the sign of the integral and reverse the limits.

Example

• If f(x) = find f’(x) 2

2

2

1x

te dt

2

2

1

2

x

tx

ddt

dx eNeither limit of integration is a constant.

2 0

0 2

1 1

2 2

x

t tx

ddt dt

dx e e

It does not matter what constant we use!

2 2

0 0

1 1

2 2

x x

t t

ddt dt

dx e e

2 2

1 12 2

22xx

xee

(Limits are reversed.)

(Chain rule is used.)2 2

2 2

22xx

x

ee

We split the integral into two parts.

HW: p. 287/37-42

The Fundamental Theorem of Calculus, Part 2

If f is continuous at every point of , and if

F is any antiderivative of f on , then

,a b

b

af x dx F b F a

,a b

(Also called the Integral Evaluation Theorem)

To evaluate an integral, take the anti-derivatives and subtract.

Antiderivatives

• Antiderivatives are also called indefinite integrals

• They are sometimes written

• Note that there are no limits on the integral

• Do not confuse with definite integrals!

F x f x dx

Common Antiderivatives

2

32

1

1)

2)2

3)3

4)1

15) ln

nn

dx x C

xxdx C

xx dx C

xx dx C

n

dx x Cx

2

2

6) sin cos

7) cos sin

8) sec tan

9) csc cot

10) sec tan sec

11) csc cot csc

xdx x C

xdx x C

xdx x C

xdx x C

x xdx x C

x xdx x C

Evaluate the integral using FTC2

35

1

8

2

42

0

1)

2) 4 3

3) 1 3

x dx

x dx

y y dy

Rewrite then evaluate the integral using FTC2

4

0

2

41

22

0

2 2

31

2 2

1

1)

32)

3) 1

44)

15)

xdx

dtt

y dy

udu

u

xdx

x

Evaluate the integral involving trigonometric functions using FTC22

32

6

2

3

24

20

1) cos

2) csc

3) csc cot

1 cos4)

cos

d

d

x xdx

udu

u

Special Example: absolute value

22

0

x x dx

Area using Integrals

• Find the zeros of the function over the interval [a,b]

• integrate over each subinterval

• add the absolute value of the integrals

Example: Find the area using integrals

21) 4 ; [0, 3]y x

Using the GC to find the integral

• hit MATH then 9

• fnInt( will come up on the screen

• type in the function, comma, x, comma, -a, comma, b) then hit ENTER

• Ex: 2

1

sin ( sin( ), , 1,2) 2.043x xdx fnINT x x x

Examples: Use GC

2

1

20

5

0

41)

1

2) x

dxx

e dx

Area using GC

• To find the area under the curve f(x) from [a,b] type fnInt(abs(f(x)),x,a,b)

• Example: Find the area under the curve

y = xcos2x on [-3, 3]

HW: FTC 2 wksheet