Antiderivatives and Indefinite Integration

Lesson 5.1

Reversing Differentiation

• An antiderivative of function f is • a ________________F• which satisfies __________

• Consider the following:

• We note that two antiderivatives of the same function differ by a __________________

4

3

( ) 2

'( ) 8

F x x

F x x

4

3

( ) 2 17

'( ) 8

G x x

G x x

Reversing Differentiation

• General antiderivativesf(x) = 6x2 F(x) = 2x3 + C• because ___________ = 6x2

k(x) = sec2(x) K(x) = ___________________• because K’(x) = k(x)

Differential Equation

• A differential equation in x and y involves• x, y, and _____________________ of y

• Examples

• Solution – find a function whose ___________is the differential given

3

' 5

' 4

y x

y x

Differential Equation

• When

• Then one such function is

• The general solution is

' 5y x

25

2y x

Notation for Antiderivatives

• We are starting with

• Change to differential form

• Then the notation for antiderivatives is

( )dy

f xdx

( ) ( )y f x dx F x C "The ______________of f with respect to x"

Basic Integration Rules

• Note the inverse nature of integration and differentiation

• Note basic rules, pg 286

'( ) ( )F x dx F x C

Practice

• Try these3 24 6 1x x dx

2

4

2 3x xdx

x

sec tan secy y y dy

Finding a Particular Solution

• Given

• Find the specific equation from the family of antiderivatives, whichcontains the point (3,2)

• Hint: find the __________________, use the given point to find the value for C

2 1dy

xdx

Assignment A

• Lesson 5.1 A• Page 291• Exercises 1 – 55 odd

Slope Fields

• Slope of a function f(x)• at a point a• given by f ‘(a)

• Suppose we know f ‘(x)• substitute different values for a • draw short slope lines for successive values of y

• Example

'( ) 2f x x

Slope Fields

• For a large portion of the graph, when

• We can trace the line for a specific F(x)• specifically when the C = -3

'( ) 2f x x

Finding an Antiderivative Using a Slope Field

• Given

• We can trace the version of the original F(x) which _______________________.

2

'( ) xf x e

Vertical Motion

• Consider the fact that the acceleration due to gravity a(t) = -32 fps

• Then v(t) = -32t + v0 • Also s(t) = -16t2 + v0t + s0

• A balloon, rising vertically with velocity = 8 releases a sandbag at the instant it is 64 feet above the ground• How long until the sandbag hits the ground• What is its velocity when this happens?

Why?Why?

Rectilinear Motion

• A particle, initially at rest, moves along the x-axis at velocity of

• At time t = 0, its position is x = 3• Find the velocity and position functions for the

particle• Find all values of t for which the particle is at rest

( ) cos 0a t t t

Assignment B

• Lesson 5.1 B• Page 292• Exercises 57 – 93, EOO

Area as the Limit of a SumLesson 5.2

Area under f(x) = ln x

• Consider the task to compute the area under a curve f(x) = ln x on interval [1,5]

1 2 3 4 5

x

We estimate with 4 rectangles using the _________endpoints

Area under the Curve

1 2 3 4 5

x

4 ln 2 ln 3 ln 4 ln 5S x x x x

We can ________________our estimate by increasing the number of rectangles

Area under the Curve• Increasing the number of rectangles to n

• This can be done on the calculator:

5 1 4b ax

n n n

4 1 2 3ln ln ln ... ln nS x x x x x x x x

Generalizing

• In general …

• The actual area is

• where

a b

( ) ... ( )nS f a x x f a n x x

_______ ( ) ... ( )f a x x f a n x x

Summation Notation

• We use summation notation

• Note the basic rules and formulas• Examples pg. 295• Theorem 5.2 Formulas, pg 296

1 21

...n

n kk

a a a a

Use of Calculator

• Note again summation capability of calculator• Syntax is:

(expression, variable, low, high)

Practice Summation

• Try these

50

1

3k

5

1

( 1)k

k

402

1

( 1)k

k

Practice Summation

• For our general formula:

• let f(x) = 3 – 2x on [0,1]

Assignment

• Lesson 5.2• Page 303• Exercises 1 – 61 EOO

(omit 45)

Riemann Sums and the Definite Integral

Lesson 5.3

Review

• We partition the interval into n ____________

• Evaluate f(x) at _________endpointsof kth sub-interval for k = 1, 2, 3, … n

a b

f(x)

b ax

n

Review

• Sum

• We expect Sn to improve thus we define A, the ______________under the curve, to equal the above limit.

a b

1

lim ( )n

nn

k

S f a k x x

f(x)

Riemann Sum

1. Partition the interval [a,b] into n subintervalsa = x0 < x1 … < xn-1< xn = b

• Call this partition P• The kth subinterval is xk = xk-1 – xk

• Largest xk is called the _________, called ||P||

2. Choose an arbitrary value from each

subinterval, call it _________

Riemann Sum3. Form the sum

This is the Riemann sum associated with• the function ______• the given partition ____• the chosen subinterval representatives ______

• We will express a variety of quantities in terms of the Riemann sum

1 1 2 21

( ) ( ) ... ( ) ( )n

n n n i ii

R f c x f c x f c x f c x

1 1 2 21

( ) ( ) ... ( ) ( )n

n n n i ii

R f c x f c x f c x f c x

The Riemann SumCalculated

• Consider the function2x2 – 7x + 5

• Use x = 0.1

• Let the = left edgeof each subinterval

• Note the sum

x 2x 2̂-7x+5 dx * f(x)4 9 0.9

4.1 9.92 0.9924.2 10.88 1.0884.3 11.88 1.1884.4 12.92 1.2924.5 14 1.44.6 15.12 1.5124.7 16.28 1.6284.8 17.48 1.7484.9 18.72 1.872

5 20 25.1 21.32 2.1325.2 22.68 2.2685.3 24.08 2.4085.4 25.52 2.5525.5 27 2.75.6 28.52 2.8525.7 30.08 3.0085.8 31.68 3.1685.9 33.32 3.332

Riemann sum = 40.04

x 2x 2̂-7x+5 dx * f(x)4 9 0.9

4.1 9.92 0.9924.2 10.88 1.0884.3 11.88 1.1884.4 12.92 1.2924.5 14 1.44.6 15.12 1.5124.7 16.28 1.6284.8 17.48 1.7484.9 18.72 1.872

5 20 25.1 21.32 2.1325.2 22.68 2.2685.3 24.08 2.4085.4 25.52 2.5525.5 27 2.75.6 28.52 2.8525.7 30.08 3.0085.8 31.68 3.1685.9 33.32 3.332

Riemann sum = 40.04

ic

The Riemann Sum

• We have summed a series of boxes• If the x were ____________________, we

would have gotten a better approximation

f(x) = 2x2 – 7x + 5

1

( ) 40.04n

i ii

f c x

The Definite Integral

• The definite integral is the _______of the Riemann sum

• We say that f is _____________ when• the number I can be approximated as accurate as

needed by making ||P|| sufficiently small• f must exist on [a,b] and the Riemann sum must

exist

0

1

( ) limnb

i ia Pk

I f x dx f c x

Example

• Try

• Use summation on calculator.

3 4

24

11

use (1 )k

x dx S f k x x

b ax

n

Example

• Note increased accuracy with __________ x

Limit of the Riemann Sum

• The definite integral is the ___________of the Riemann sum.

3

2

1

x dx

Properties of Definite Integral

• Integral of a sum = sum of integrals• Factor out a _________________• Dominance

( ) ( ) [ , ]

( ) ( )b b

a a

f x g x on a b

f x dx g x dx

Properties of Definite Integral

• Subdivision rule

( ) ( ) ____________c b

a a

f x dx f x dx

a b c

f(x)

Area As An Integral

• The area under the curve on theinterval [a,b] a c

f(x)

A

Distance As An Integral

• Given that v(t) = the velocity function with respect to time:

• Then _____________________ can be determined by a definite integral

• Think of a summation for many small time slices of distance

( )t b

t a

D v t dt

Assignment

• Section 5.3• Page 314• Problems: 3 – 49 odd

The Fundamental Theorems of Calculus

Lesson 5.4

First Fundamental Theorem of Calculus

• Given f is • _________________on interval [a, b]• F is any function that satisfies F’(x) = f(x)

• Then

( ) __________________b

af x dx

First Fundamental Theorem of Calculus

• The definite integral

can be computed by• finding an _________________F on interval [a,b]• evaluating at limits a and b and _____________

• Try

( )b

af x dx

7

36x dx

Area Under a Curve

• Consider

• Area =

sin cos on 0,2

y x x

Area Under a Curve

• Find the area under the following function on the interval [1, 4]

2( 1)y x x x

Second Fundamental Theorem of Calculus

• Often useful to think of the following form

• We can consider this to be a _______________ in terms of x

( )x

af t dt

( ) ( )x

aF x f t dt View QuickTime

Movie

View QuickTime Movie

Second Fundamental Theorem of Calculus• Suppose we are

given G(x)

• What is G’(x)?

4( ) (3 5)

xG x t dt

Second Fundamental Theorem of Calculus

• Note that

• Then

• What about ?

( ) ( )

( ) ( )

x

aF x f t dt

F x F a

( ) ( )a

xF x f t dt

Since this is a _____________

Since this is a _____________

Second Fundamental Theorem of Calculus• Try this 1

2( )

1 3x

dtF x dt

t

( ) ( )

( ) ( )

so

a

xF x f t dt

F a F x

Assignment

• Lesson 5.4• Page 327• Exercises 1 – 49 odd

Integration by SubstitutionLesson 5.5

Substitution with Indefinite Integration

• This is the “backwards” version of the _____________________

• Recall …

• Then …

5 42 24 7 5 4 7 2 4dx x x x x

dx

4 52 25 4 7 2 4 4 7x x x dx x x C

Substitution with Indefinite Integration

• In general we look at the f(x) and “split” it• into a ________________________

• So that …

( )f x dx

( ) ( )du

f x g udx

( ) ( )du

g u dx G u Cdx

Substitution with Indefinite Integration

• Note the parts of the integral from our example

( ) ( )du

g u dx G u Cdx

4 52 25 4 7 2 4 4 7x x x dx x x C

Example

• Try this … • what is the g(u)?• what is the du/dx?

• We have a problem …

2(4 5)x dx

Where is the 4 which we need?Where is the 4 which we need?

Example

• We can use one of the properties of integrals

• We will insert a factor of _____inside and a factor of ¼ __________to balance the result

2)44

(41

5x dx

( ) ( )c f x dx c f x dx

Can You Tell?

• Which one needs substitution for integration?

• Go ahead and do the integration.

2

52

3 5

3 5

x x dx

x x dx

Try Another …

3 1t dt3sin cosx x dx

Assignment A

• Lesson 5.5• Page 340• Problems:

1 – 33 EOO49 – 77 EOO

Change of Variables

• We completely rewrite the integral in terms of u and du

• Example:

• So u = _________ and du = _________• But we have an x in the integrand

• So we solve for x in terms of u

2 3x x dx

3

2

ux

Change of Variables

• We end up with

• It remains to distribute the and proceed with the integration

• Do not forget to "_________________"

2 3 ________________x x dx 1

2u

What About Definite Integrals

• Consider a variationof integral from previous slide

• One option is to change the limits• u = __________ Then when t = 1, u = ___

when t = 2, u = ____• Resulting integral

2

1

3 1t dt

What About Definite Integrals

• Also possible to "un-substitute" and use the ___________________ limits

21 3 3

2 2 2

1

1 1 2 23 1

3 3 3 9u du u t

Integration of Even & Odd Functions

• Recall that for an even function• The function is symmetric about the ________

• Thus

• An odd function has• The function is symmetric about the orgin

• Thus

( ) ( )f x f x

0

( ) 2 ( )a a

a

f x dx f x dx

( ) _______a

a

f x dx

Assignment B

• Lesson 5.5• Page 341• Problems:

87 - 109 EOO117 – 132 EOO

Numerical Integration

Lesson 5.6

Trapezoidal Rule

• Instead of calculatingapproximation rectangleswe will use trapezoids• More accuracy

• Area of a trapezoid

a bx

•ix

b1

b2

h____________A

• Which dimension is the h?

• Which is the b1 and the b2

• Which dimension is the h?

• Which is the b1 and the b2

Trapezoidal Rule

• Trapezoidal rule approximates the integral

• Calculator function for f(x)((2*f(a+k*(b-a)/n),k,1,n-1)+f(a)+f(b))*(b-a)/(n*2)trap(a,b,n)

dx

f(xi)f(xi-1)

0 1 2 1( ) ( ) 2 ( ) 2 ( ) ...2 ( ) ( )2

where ____________

b

n n

a

dxf x dx f x f x f x f x f x

dx

Trapezoidal Rule• Entering the trapezoidal rule into the

calculator

• __________ must be defined for this to work

Trapezoidal Rule

• Try using the trapezoidal rule

• Check with integration

25

0

2 8x dx n

Simpson's Rule• As before, we divide

the interval into n parts• n must be ___________

• Instead of straight lines wedraw _____________through each group of three consecutive points• This approximates the original curve for finding

definite integral – formula shown below

a b•ix

0 1 2 3 4

2 1

( ) [ ( ) 4 ( ) 2 ( ) 4 ( ) 2 ( )3

... 2 ( ) 4 ( ) ( )]

b

a

n n n

dxf x dx f x f x f x f x f x

f x f x f x

Simpson's Rule

• Our calculator can do this for us also• The function is more than a one liner

• We will use the program editor• Choose APPS,

7:Program Editor3:New

• Specify Function,name it simp

Simpson's Rule

• Enter the parameters a, b, and n between the parentheses

Enter commands shown between Func and endFunc

Simpson's Rule

• Specify a function for ______________• When you call simp(a,b,n),

• Make sure n is an number

• Note the accuracy of the approximation

Assignment A

• Lesson 5.6• Page 350• Exercises 1 – 23 odd

Error Estimation

• Trapezoidal error for f on [a, b]• Where M = _______________of |f ''(x)| on [a, b]

• Simpson's errorfor f on [a, b]• Where K = max value of ___________ on [a, b]

3

212n

b aE M

n

5

4180n

b aE K

n

Using Data

• Given table of data, use trapezoidal rule to determine area under the curve

• dx = ?

x 2.00 2.10 2.20 2.30 2.40 2.50 2.60

y 4.32 4.57 5.14 5.78 6.84 6.62 6.51

0 1 2 1( ) ( ) 2 ( ) 2 ( ) ...2 ( ) ( )2

b

n n

a

dxf x dx f x f x f x f x f x

Using Data

• Given table of data, use Simpson's rule to determine area under the curve

0 1 2 3 4

2 1

( ) [ ( ) 4 ( ) 2 ( ) 4 ( ) 2 ( )3

... 2 ( ) 4 ( ) ( )]

b

a

n n n

dxf x dx f x f x f x f x f x

f x f x f x

x 2.00 2.10 2.20 2.30 2.40 2.50 2.60

y 4.32 4.57 5.14 5.78 6.84 6.62 6.51

Assignment B

• Lesson 5.6• Page 350• Exercises 27 – 39 odd

49, 51, 53

The Natural Log Function: Integration

Lesson 5.7

Log Rule for Integration

• Because

• Then we know that

• And in general, when u is a differentiable function in x:

1ln( )

dx

dx x

1dxx

Try It Out

• Consider these . . .

2

33

xdx

x

2sec

tan

xdx

x

Finding Area

• Given

• Determine the area under the curve on the interval [2, 4]

2

lny

x x

Using Long Division Before Integrating

• Use of the log rule is often in disguised form

• Do the division on this integrand and alter it's appearance

22 7 3

2

x xdx

x

22 2 7 3x x x

Using Long Division Before Integrating

• Calculator also can be used

• Now take the integral

192 11

2x dx

x

Change of Variables

• Consider

• Then u = x – 1 and du = dx• But x = _________ and x – 2 = ______________

• So we have

• Finish the integration

3

2

1

x xdx

x

Integrals of Trig Functions

• Note the table of integrals, pg 357• Use these to do integrals involving trig

functions

tan 5 d

1

0

sin cost t dt

Assignment

• Assignment 5.7• Page 358• Exercises 1 – 37 odd

69, 71, 73

Inverse Trigonometric Functions: Integration

Lesson 5.8

Review

• Recall derivatives of inverse trig functions

92

1

2

12

1

2

1sin , 1

11

tan1

1sec , 1

1

d duu u

dx dxud du

udx u dxd du

u udx dxu u

Integrals Using Same Relationships

93

2 2

2 2

2 2

_____________

1_____________

1___________

du uC

aa udu u

Ca u a adu u

Ca au u a

When given integral problems, look for

these patterns

When given integral problems, look for

these patterns

Identifying Patterns

• For each of the integrals below, which inverse trig function is involved?

94

2

4

13 16

dx

x 225 4

dx

x x

29

dx

x

Warning

• Many integrals look like the inverse trig forms• Which of the following are of the inverse trig

forms?

95

21

dx

x

21

x dx

x

21

dx

x

21

x dx

x

If they are not, how are they integrated?

If they are not, how are they integrated?

Try These

• Look for the pattern or how the expression can be manipulated into one of the patterns

96

2

8

1 16

dx

x

21 25

x dx

x

24 4 15

dx

x x

2

5

10 16

xdx

x x

Completing the Square

• Often a good strategy when quadratic functions are involved in the integration

• Remember … we seek _______________• Which might give us an integral resulting in the

arctan function

2 2 10

dx

x x

Completing the Square

• Try these2

22 4 13

dx

x x

2

2

4dx

x x

Rewriting as Sum of Two Quotients

• The integral may not appear to fit basic integration formulas• May be possible to ______________________into

two portions, each more easily handled

2

4 3

1

xdx

x

Basic Integration Rules

• Note table of basic rules• Page 364

• Most of these should be committed to memory

• Note that to apply these, you must create the proper ________ to correspond to the u in the formula

cos sinu du u C

Assignment

• Lesson 5.8• Page 366• Exercises 1 – 39 odd

63, 67

101

Hyperbolic Functions -- Lesson 5.9

• Consider the following definitions

• Match the graphs with the definitions.• Note the identities, pg. 371

sinh2

cosh2

sinhtanh

cosh

x x

x x

e ex

e ex

x

x

Derivatives of Hyperbolic Functions

• Use definitions to determine the derivatives• Note the pattern or interesting results

sinh sinh ??2

cosh cosh ??2

sinhtanh tanh ??

cosh

x x

x x

e e dx x

dx

e e dx x

dxx d

xx dx

Integrals of Hyperbolic Functions

• This gives us antiderivatives (integrals) of these functions

• Note other derivatives, integrals, pg. 371

2

cosh ____________

sinh cosh

sech __________

u du C

u du u C

u du C

Integrals Involving Inverse Hyperbolic Functions

105

1

2 2

1

2 2

-12 2

1

2 2

1sinh

1cosh

1 1tanh

1 1sech

udu C

au au

du Cau au

du Ca u a a

udu C

a au a u

Try It!

• Note the definite integral

• What is the a, the u, the du?• a = 3, u = _________, du = _______________

106

4

21

1

9 4dx

x

4

21

1 2

2 9 4dx

x

Application

• Find the area enclosed by x = -¼, x = ¼, y = 0, and

• Which pattern does this match?• What is the a, the u, the du?

107

2

1

1 4y

x

Assignment

• Lesson 5.9• Page 377• Exercises 1 – 29 EOO

37 – 53 EOO

108