John Zhu

Improper Integrals

Slide Duration:

Table of Contents

Section 1: Prerequisites

Parametric Curves

10m 10s

Intro

0:00

Parametric Equations

0:23

Familiar Functions

0:32

Parametric Equation/ Function

0:56

Example 1: Graph Parametric Equation

1:48

Example 2

4:30

Example 3

6:01

Example 4

7:12

Example 5

8:10

Polar Coordinates

14m 54s

Intro

0:00

Polar Coordinates

0:09

Definition

0:11

Example

0:23

Converting Polar Coordinates

1:49

Example: Convert Polar Equation to Cartesian Coordinates

3:06

Example: Convert Polar Equation to Cartesian Coordinates

5:21

Example: Find the Polar Representation

6:51

Example: Find the Polar Representation

8:39

Example: Sketch the Graph

10:02

Vector Functions

12m 5s

Intro

0:00

Vector Functions

0:11

2 Parts

0:18

Example 1

1:44

Example 2

3:52

Example 3

4:59

Example 4

6:03

Example 5

8:00

Example 6

9:23

Example 7

10:16

Section 2: Differentiation

Parametric Differentiation

13m 15s

Intro

0:00

Parametric Differentiation

0:15

Example 1

1:16

Example 2

1:54

Example 3

3:15

Example 4

4:59

Parametric Differentiation: Position, Speed & Acceleration

7:45

Example 5

8:32

Example 6

10:37

Polar Differentiation

10m 14s

Intro

0:00

Polar Differentiation

0:11

Goal

0:13

Method

0:24

Example 1

0:52

Example 2

4:27

Example 3

7:03

L'Hopital's Rule

6m 43s

Intro

0:00

L'Hopitals Rule

0:09

Conditions

0:14

Limit

0:36

Example 1

1:21

Example 2

2:38

Example 3

4:09

Example 4

5:01

Section 3: Integration

Integration By Parts

19m 4s

Intro

0:00

Integration by Parts

0:12

Formula

0:31

How It's Derived From Product Rule

1:09

Integration by Parts: Rules to Follow

3:24

Recap of the Rules

4:13

Example 1

4:29

Example 2

7:04

Example 3

8:40

Example 4

13:48

Example 5

15:36

Integration By Partial Fractions

22m 4s

Intro

0:00

Integration by Partial Fractions: Goal

0:08

Example 1

2:10

Example 2

7:10

Example 3

10:06

Example 4

14:23

Example 5

16:26

Improper Integrals

19m 1s

Intro

0:00

Improper Integrals

0:06

3 Steps

1:11

Example 1

1:46

Example 2

3:20

Example 3

6:05

Example 4

9:02

Example 5

11:21

Example 6

15:54

Section 4: Applications of Integration

Logistic Growth

19m 16s

Intro

0:00

Logistic Growth Function

0:07

Defining Variables

0:40

Equation Parts

1:51

Logistic Growth

2:04

Example 1

2:59

Example 2

7:13

Example 3

11:29

Example 4

15:21

Arc Length for Parametric & Polar Curves

17m 40s

Intro

0:00

Arc Length

0:13

Arc Length of a Normal Function

0:24

Example

1:27

Example 2: Arc Length for Parametric Curves

3:31

Example 3: Arc Length for Parametric Curves

4:23

Example 4: Arc Length for Parametric Curves

8:05

Example 5: Arc Length for Parametric Curves

12:22

Example 6: Arc Length for Polar Curves

15:36

Example 7: Arc Length for Polar Curves

16:03

Area for Parametric & Polar Curves

4m 33s

Intro

0:00

Area for Parametric Curves: Parametric Function

0:10

Example 1: Area for Parametric Curves

0:35

Area for Parametric Curves: Polar Function

2:50

Example 2: Area for Polar Curves

3:18

Section 5: Sequences, Series, & Approximations

Definition & Convergence

7m 10s

Intro

0:00

Sequences: Definition

0:09

Definition

0:31

Example 1

1:07

Sequences: Convergence

2:02

Example 1

2:52

Example 2

3:36

Example 3

4:47

Example 4

6:16

Geometric Series

10m 59s

Intro

0:00

Geometric Series

0:07

Definition

0:24

Expanded Form

0:39

Convergence Rules

1:00

Example 1: Convergence

1:22

Example 2: Convergence

2:36

Example 3: Convergence

3:45

Sum of Series

5:04

Sum of First n Terms

5:14

Sum of Series

5:24

Example 1: Sum of Series

5:39

Example 2: Sum of Series

7:15

Example 3: Sum of Series

8:24

Example 4: Sum of Series

9:36

Harmonic & P Series

3m 58s

Intro

0:00

Harmonic Series

0:08

P-Series

1:17

Example 1: P-Series Test

2:22

Example 2: P-Series Test

3:06

Integral Test

11m 31s

Intro

0:00

Integral Test

0:09

Example 1

0:54

Example 2

4:00

Example 3

7:59

Example 4

9:47

Comparison Test

7m 47s

Intro

0:00

Comparison Test

0:10

Example 1

1:07

Example 2

2:33

Example 3

4:20

Example 4

6:29

Limit Comparison Test

9m 40s

Intro

0:00

Limit Comparison Test

0:09

Conditions

0:31

Example 1

1:01

Example 2

2:53

Example 3

4:15

Example 4

6:19

Ratio Test

11m 20s

Intro

0:00

Ratio Test

0:09

Example 1

0:57

Example 2

2:55

Example 3

6:27

Example 4

8:36

Alternating Series

9m 11s

Intro

0:00

Alternating Series

0:08

Convergence Test

0:59

Example 1

1:27

Example 2

2:42

Example 3

4:57

Example 4

6:37

Absolute Convergence

7m 13s

Intro

0:00

Absolute Convergence

0:12

Example 1

0:52

Example 2

3:42

Example 3

5:21

Power Series Convergence

12m 52s

Intro

0:00

Power Series

0:09

Definition

0:19

Radius & Interval of Convergence

2:07

Example 1

2:28

Example 2

4:18

Example 3

6:20

Example 4

9:11

Taylor Series

15m 11s

Intro

0:00

Taylor Series

0:06

MacLaurin Series

0:45

Example 1

1:02

Example 2

2:45

Example 3

6:04

Example 4

8:23

Example 5

11:49

Power Series Operations

16m 40s

Intro

0:00

Operations

0:07

Example 1: Substitution

1:05

Example 2: Substitution

3:41

Example 3: Differentiation/ Integration

5:39

Example 4: Differentiation/ Integration

9:55

Example 5: Multiplying/ Dividing

12:32

Example 6: Multiplying/ Dividing

14:50

Lagrange Error

7m 9s

Intro

0:00

Power Series: Lagrange Error

0:06

Lagrange Remainder

0:21

Lagrange Error Bound

0:50

Example 1

1:06

Example 2

3:27

Section 6: Practice Test

AP Calc BC Practice Test Section 1: Multi Choice Part 1

15m 45s

Intro

0:00

Practice Problem 1

0:10

Practice Problem 2

1:19

Practice Problem 3

2:33

Practice Problem 4

5:25

Practice Problem 5

6:32

Practice Problem 6

9:11

Practice Problem 7

11:31

Practice Problem 8

13:08

Practice Problem 9

14:16

AP Calc BC Practice Test Section 1: Multi Choice Part 2

21m 38s

Intro

0:00

Practice Problem 10

0:12

Practice Problem 11

3:03

Practice Problem 12

4:53

Practice Problem 13

6:56

Practice Problem 14

9:10

Practice Problem 15

12:17

Practice Problem 16

14:43

Practice Problem 17

16:18

Practice Problem 18

18:34

AP Calc BC Practice Test Section 1: Multi Choice Part 3

17m 31s

Intro

0:00

Practice Problem 19

0:09

Practice Problem 20

3:03

Practice Problem 21

4:07

Practice Problem 22

5:41

Practice Problem 23

8:48

Practice Problem 24

11:16

Practice Problem 25

12:53

Practice Problem 26

14:12

Practice Problem 27

15:21

AP Calc BC Practice Test Section 1: Multi Choice Part 4

14m 17s

Intro

0:00

Practice Problem 28

0:08

Practice Problem 29

1:19

Practice Problem 30

2:15

Practice Problem 31

4:19

Practice Problem 32

6:54

Practice Problem 33

8:13

Practice Problem 34

9:03

Practice Problem 35

10:21

Practice Problem 36

11:42

AP Calc BC Practice Test Section 1: Multi Choice Part 5

22m 45s

Intro

0:00

Practice Problem 37

0:10

Practice Problem 38

4:09

Practice Problem 39

7:21

Practice Problem 40

9:42

Practice Problem 41

11:35

Practice Problem 42

14:01

Practice Problem 43

15:50

Practice Problem 44

16:43

Practice Problem 45

19:35

AP Calc BC Practice Test Section 1: Free Response Part 1

9m 55s

Intro

0:00

Practice Problem 1a

0:08

Practice Problem 1b

1:30

Practice Problem 1c

3:08

Practice Problem 2a

3:50

Practice Problem 2b

5:40

Practice Problem 2c

7:24

AP Calc BC Practice Test Section 1: Free Response Part 2

10m 8s

Intro

0:00

Practice Problem 3a

0:09

Practice Problem 3b

3:01

Practice Problem 4a

3:55

Practice Problem 4b

5:43

AP Calc BC Practice Test Section 1: Free Response Part 3

13m 58s

Intro

0:00

Practice Problem 5a

0:07

Practice Problem 5b

2:21

Practice Problem 5c

4:21

Practice Problem 6a

6:16

Practice Problem 6b

8:57

Practice Problem 6c

11:03

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Calculus BC Improper Integrals

Section 3: Integration: Lecture 3 | 19:01 min

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Lecture Comments (8)

	0 answers Post by Kevin Hong on April 27, 2014 at about 5:45, you said inverse tangent of infinity is pi over 2 but how did you come up with pi over 2?
	1 answer Last reply by: Huijie Shen Sun Apr 19, 2015 8:55 PM Post by Mohamad Oda on April 6, 2014 2*sqrt(2) is the right answer
	2 answers Last reply by: Thomas Zhang Mon Mar 17, 2014 1:10 PM Post by Raudel Ulloa on August 6, 2012 I checked in the calcuator for number 5 and the interagal is 2 radical 2, what i got, not 4
	1 answer Last reply by: Raudel Ulloa Mon Aug 6, 2012 9:48 AM Post by Chung Teak Joon on April 29, 2012 isnt progress on number four kind of wrong? or am I just misunderstanding the steps? If we integrate root u, doesnt it have to become 0.5u^0.5??? How did it suddenly become 2u^0.5????

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Working on the solution...

Improper Integrals

Use method of improper integrals when:

There is an infinity in either bound of the integral
Function evaluated at any bound is undefined

Improper Integrals

∫₁^∞ [3/x]dx

Replace the improper bound ∞ with k
∫₁^k [3/x]dx
Integrate
∫₁^k [3/x]dx = 3[ln|x|]₁^k
= 3lnk − 3ln1

= 3lnk = lim_{k → ∞} 3lnk = ∞

∫_{− ∞}⁰ [1/(( x + 5 )²)]dx

Replace the improperbound ∞ with k
∫_k⁰ [1/(( x + 5 )²)]dx
Integrate
∫_k⁰ [1/(( x + 5 )²)]dx = [ − [1/(x + 5)] ]_k^0.
= − [1/(0 + 5)] + [1/(k + 5)]

= lim_{x → − ∞} [ [1/(k + 5)] ] − [1/5] = 0 − [1/5] = − [1/5]

∫_{− ∞}^∞ [7/((x − 3)²)]dx

Note that [7/((x − 3)²)] is an even function,and the limits are the same magnitude
∫_{− ∞}^∞ [7/((x − 3)²)]dx = 2∫₀^∞ [7/((x − 3)²)]dx
Replace the improper limit ∞ with k
2∫₀^∞ [7/((x − 3)²)]dx = 2∫₀^k [7/((x − 3)²)]dx
Integrate
2∫₀^k [7/((x − 3)²)]dx = 14∫₀^k [1/((x − 3)²)]dx
= 14[ − [1/(x − 3)] ]₀^k
= 14[ − [1/(k − 3)] + [1/(0 − 3)] ]
= 14[ − [1/(k − 3)] − [1/3] ]
= 14[ − lim_{k → ∞} [1/(k − 3)] − [1/3] ]
= 14[ 0 − [1/3] ]

= [( − 14)/3]

∫₃⁷− [4/(√{x − 3} )]dx

Replace the bound 3 with k
∫₃⁷− [4/(√{x − 3} )]dx = ∫_k⁷− [4/(√{x − 3} )]dx
Integrate
∫_k⁷− [4/(√{x − 3} )]dx = − 4∫_k⁷ [1/(√{x − 3} )]dx
= − 4[ 2√{x − 3} ]_k⁷
= − 4[2√{7 − 3} − 2√{k − 3} ]
= − 4[2√4 − 2√{k − 3} ]

= − 4[4 − 0] = − 4[4] = − 16

∫_{− 2}^k [2/(x√{x² − 1} )]dx

Replace the bound 1 with k
∫_{− 2}^k [2/(x√{x² − 1} )]dx = ∫_{− 2}^k [2/(x√{x² − 1} )]dx
Integrate using inverse trigonmetric identity
∫_{− 2}^k [2/(x√{x² − 1} )]dx = 2∫_{− 2}^k [1/(x√{x² − 1} )]dx
= 2[ sec^{− 1}|x| ]_{− 2}^k
= 2[ sec^{− 1}|k| − sec^{− 1}| − 2| ]

= 2[ lim_k1 sec^{− 1}|k| − [(π)/3] ] = 2[ 0 − [(π)/3] ] = − [(2π)/3]

∫₀^∞ [(x − 4)/(x² − 3x − 4)]dx

Factor x² − 3x − 4 to simplify
∫₀^∞ [(x − 4)/(x² − 3x − 4)]dx = ∫₀^∞ [(x − 4)/((x − 4)(x + 1))]dx
= ∫₀^∞ [1/((x + 1))]dx
Replace the improper limit ∞ with k
∫₀^∞ [1/((x + 1))]dx = ∫₀^k [1/((x + 1))]dx
Integrate
∫₀^k [1/((x + 1))]dx = [ ln|x + 1| ]₀^k
= ln|k + 1| − ln|0 + 1|
= ln|k + 1| − ln|1|
= ln|k + 1| − 0
= ln|k + 1|
= lim_{k → ∞} ln|k + 1|

= ∞

∫_{− ∞}¹ [2/(x² + 2x)]dx

Replace the bound − ∞ with k
∫_{− ∞}¹ [2/(x² + 2x)]dx = ∫_k¹ [2/(x² + 2x)]dx
Integrate
∫_k¹ [2/(x² + 2x)]dx = ∫_k¹ [1/x] − [1/(x + 2)]dx
= ∫_k¹ [1/x] − ∫_k¹ [1/(x + 2)]dx
= [ ln|x| ]_k¹ − [ ln|x + 2| ]_k¹
= ln1 − lnk − ln3 + ln|k + 2|

= lim_{k → − ∞} [ − ln3 + ln|k + 2| − lnk ] = − ln3 − ∞− ∞ = − ln3

∫₀^∞ [4/((x + 1)(x + 2))] dx

Replace the bound ∞ with k
∫₀^∞ [4/((x + 1)(x + 2))] dx = ∫₀^k [4/((x + 1)(x + 2))] dx
Integrate
∫₀^k [4/((x + 1)(x + 2))] dx = ∫₀^k [4/(x + 1)] − [4/(x + 2)] dx
= 4∫₀^k [1/(x + 1)] − [1/(x + 2)] dx
= 4[ ln|x + 1| − ln|x + 2| ]₀^k

= 4[ lim_{k → ∞} ln[(k + 1)/(k + 2)] − ln[1/2] ] = 4[ 0 − ln[1/2] ] = − 4ln[1/2]

∫_{− 2}² [1/(√{4 − x²} )]dx

Note the function is symmetrical and simplify
∫_{− 2}² [1/(√{4 − x²} )]dx = 2∫₀² [1/(√{4 − x²} )]dx
Replace the bound 2 with k
2∫₀² [1/(√{4 − x²} )]dx = 2∫₀^k [1/(√{4 − x²} )]dx
Apply the inverse trignometric integration identity to integrate
2∫₀^k [1/(√{4 − x²} )]dx = 2[ sin^{− 1}[x/2] ]₀^k
= 2[ sin^{− 1}[k/2] − sin^{− 1}[0/2] ]
= 2[ sin^{− 1}[k/2] ]
= 2[ lim_{k → 2} sin^{− 1}[k/2] ]
= 2[ [(π)/2] ]

= π

10.∫₀^∞ [1/(( x − 2 )³)] dx

Replace the bound ∞ with k
∫₀^∞ [1/(( x − 2 )³)] dx = ∫₀^k [1/(( x − 2 )³)] dx
Integrate
∫₀^k [1/(( x − 2 )³)] dx = [ − [1/(2(x − 2)²)] ]₀^k
= [ − lim_{k → ∞} [1/(2(k − 2)²)] + [1/(2(0 − 2)²)] ]
= 0 + [1/2(4)]

= [1/8]

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Answer

Improper Integrals

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Intro

Improper Integrals

3 Steps

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Intro 0:00
Improper Integrals 0:06

3 Steps

Example 1 1:46
Example 2 3:20
Example 3 6:05
Example 4 9:02
Example 5 11:21
Example 6 15:54

Mathematics: AP Calculus BC

Section 1: Prerequisites
	Parametric Curves	10:10
	Polar Coordinates	14:54
	Vector Functions	12:05
Section 2: Differentiation
	Parametric Differentiation	13:15
	Polar Differentiation	10:14
	L'Hopital's Rule	6:43
Section 3: Integration
	Integration By Parts	19:04
	Integration By Partial Fractions	22:04
	Improper Integrals	19:01
Section 4: Applications of Integration
	Logistic Growth	19:16
	Arc Length for Parametric & Polar Curves	17:40
	Area for Parametric & Polar Curves	4:33
Section 5: Sequences, Series, & Approximations
	Definition & Convergence	7:10
	Geometric Series	10:59
	Harmonic & P Series	3:58
	Integral Test	11:31
	Comparison Test	7:47
	Limit Comparison Test	9:40
	Ratio Test	11:20
	Alternating Series	9:11
	Absolute Convergence	7:13
	Power Series Convergence	12:52
	Taylor Series	15:11
	Power Series Operations	16:40
	Lagrange Error	7:09
Section 6: Practice Test
	AP Calc BC Practice Test Section 1: Multi Choice Part 1	15:45
	AP Calc BC Practice Test Section 1: Multi Choice Part 2	21:38
	AP Calc BC Practice Test Section 1: Multi Choice Part 3	17:31
	AP Calc BC Practice Test Section 1: Multi Choice Part 4	14:17
	AP Calc BC Practice Test Section 1: Multi Choice Part 5	22:45
	AP Calc BC Practice Test Section 1: Free Response Part 1	9:55
	AP Calc BC Practice Test Section 1: Free Response Part 2	10:08
	AP Calc BC Practice Test Section 1: Free Response Part 3	13:58

Related Books

	Cracking the AP Calculus BC Exam, 2015 Edition Author: Princeton Review ISBN: 804124825 Publisher: Princeton Review Year: 2014 The book includes 3 full length practice tests with detailed explanations, a review of all the key concepts, and targeted strateeies to ace th exam for your highest score.
	Cliffs AP: Calculus AB & BC, 3rd Edition Authors: Dale W Johnson, Kerry J King ISBN: 764586831 Publisher: Cliffs Notes Year: 2001 The book has a topic-by topic breakdown and lots of problem approach suggestions for both free response and multiple choice calculus questions. The book also includes two full length tests.

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