Math 111
Calculus II
OPEN EDUCATIONAL RESOURCE
Robert Petry, Fotini Labropulu, and Iqbal Husain
LUTHER
COLLEGE
UNIVERSITY OF REGINA
published by Campion College and Luther College
,3rd Edition Copyright © 2024 Robert G. Petry, Fotini Labropulu, Iqbal Husain.
2nd Edition Copyright © 2021 Robert G. Petry, Fotini Labropulu, Iqbal Husain.
1st Edition Copyright © 2020 Robert G. Petry, Fotini Labropulu.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free
Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no
Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the
section entitled “GNU Free Documentation License”.
Permission is granted to retain (if desired) the original title of this document on modified copies.
History
• 1st Edition produced in 2021 entitled “Calculus II” written by Robert G. Petry, Fotini Labropulu, and
Iqbal Husain. Published by Campion College and Luther College.
• 2nd Edition produced in 2020 entitled “Calculus II” written by authors Robert G. Petry and Fotini
Labropulu. Published by Campion College and Luther College.
• 3rd Edition produced in 2024 entitled “Calculus II” written by authors Robert G. Petry, Fotini Labropulu,
and Iqbal Husain. Published by Campion College and Luther College. The authors gratefully acknowledge
funding provided by the University of Regina Open Education and Publishing (OEP) program that
assisted in the typesetting of this edition.
The source of this document (i.e. a transparent copy) is available via
http://campioncollege.ca/resources/dr-robert-petry
About the cover: The cover line fractal drawing is in the public domain and available from
http://openclipart.org.
, 2.5.2 Inverse Cosine . . . . . . . 47
2.5.3 Inverse Tangent . . . . . . 48
2.5.4 Other Trigonometric Inverses 49
2.6 L’Hôpital’s Rule . . . . . . . . . . 54
2.6.1 Indeterminate Forms of
Contents type 0 · ∞ and ∞ − ∞ . .
2.6.2 Exponential Indeterminate
56
Forms . . . . . . . . . . . 58
Review Exercises . . . . . . . . . . . . . . 61
3 Integration Methods 63
1 Integration Review 1 3.1 Integration by Parts . . . . . . . . 64
1.1 The Meaning of the Definite Integral 2 3.2 Trigonometric Integrals . . . . . . 68
1.2 The Fundamental Theorem of 3.3 Trigonometric Substitution . . . . 74
Calculus . . . . . . . . . . . . . . 3 3.4 Partial Fraction Decomposition . 78
1.3 Indefinite Integrals . . . . . . . . 3 3.5 General Strategies for Integration 89
1.4 Integration by Substitution . . . . 4 3.6 Improper Integrals . . . . . . . . 92
1.5 Integration Examples . . . . . . . 4 3.6.1 Improper Integrals of the
Review Exercises . . . . . . . . . . . . . 7 First Kind . . . . . . . . . 92
3.6.2 Improper Integrals of the
2 Inverses and Other Functions 9 Second Kind . . . . . . . .
95
2.1 Inverse Functions . . . . . . . . . 10 Review Exercises . . . . . . . . . . . . .
100
2.1.1 Horizontal Line Test . . . 10
2.1.2 Finding Inverse Functions 13 4 Sequences and Series 101
2.1.3 Graphs of Inverse Functions 15 4.1 Sequences . . . . . . . . . . . . . 102
2.1.4 Derivative of an Inverse 4.2 Series . . . . . . . . . . . . . . . . . 111
Function . . . . . . . . . . 15 4.3 Testing Series with Positive Terms 118
2.1.5 Creating Invertible Functions 18 4.3.1 The Integral Test . . . . . 118
2.2 Exponential Functions . . . . . . 19 4.3.2 The Basic Comparison Test 123
2.2.1 The Natural Exponential 4.3.3 The Limit Comparison Test 125
Function . . . . . . . . . . 20 4.4 The Alternating Series Test . . . 129
2.2.2 Derivative of ex . . . . . . 22 4.5 Tests of Absolute Convergence . . 132
2.2.3 Integral of ex . . . . . . . 23 4.5.1 Absolute Convergence . . 132
2.2.4 Simplifying Exponential 4.5.2 The Ratio Test . . . . . . 134
Expressions . . . . . . . . 24 4.5.3 The Root Test . . . . . . . 135
2.3 Logarithmic Functions . . . . . . 26 4.5.4 Rearrangement of Series . 137
2.3.1 Logarithmic Function 4.6 Procedure for Testing Series . . . 138
Properties . . . . . . . . . 27 4.7 Power Series . . . . . . . . . . . . 143
2.3.2 The Natural Logarithmic
4.8 Representing Functions with
Function . . . . . . . . . . 28
Power Series . . . . . . . . . . . . . 151
2.3.3 Solving Exponential and
4.9 Maclaurin Series . . . . . . . . . . 157
Logarithmic Equations . . 29
4.10 Taylor Series . . . . . . . . . . . . . 161
2.3.4 Derivative of the Natural
Review Exercises . . . . . . . . . . . . . 166
Logarithmic Function . . . 32
2.3.5 Derivatives Using Answers 169
Arbitrary Bases . . . . . . 33
2.3.6 Logarithmic Differentiation 34 Summary 183
2.3.7 Integral of x1 and ax . . . 36
2.4 Exponential Growth and Decay . 40 Index 185
2.5 Inverse Trigonometric Functions . 45
2.5.1 Inverse Sine . . . . . . . . 45 GNU Free Documentation License 187
iii
, iv
Calculus II
OPEN EDUCATIONAL RESOURCE
Robert Petry, Fotini Labropulu, and Iqbal Husain
LUTHER
COLLEGE
UNIVERSITY OF REGINA
published by Campion College and Luther College
,3rd Edition Copyright © 2024 Robert G. Petry, Fotini Labropulu, Iqbal Husain.
2nd Edition Copyright © 2021 Robert G. Petry, Fotini Labropulu, Iqbal Husain.
1st Edition Copyright © 2020 Robert G. Petry, Fotini Labropulu.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free
Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no
Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the
section entitled “GNU Free Documentation License”.
Permission is granted to retain (if desired) the original title of this document on modified copies.
History
• 1st Edition produced in 2021 entitled “Calculus II” written by Robert G. Petry, Fotini Labropulu, and
Iqbal Husain. Published by Campion College and Luther College.
• 2nd Edition produced in 2020 entitled “Calculus II” written by authors Robert G. Petry and Fotini
Labropulu. Published by Campion College and Luther College.
• 3rd Edition produced in 2024 entitled “Calculus II” written by authors Robert G. Petry, Fotini Labropulu,
and Iqbal Husain. Published by Campion College and Luther College. The authors gratefully acknowledge
funding provided by the University of Regina Open Education and Publishing (OEP) program that
assisted in the typesetting of this edition.
The source of this document (i.e. a transparent copy) is available via
http://campioncollege.ca/resources/dr-robert-petry
About the cover: The cover line fractal drawing is in the public domain and available from
http://openclipart.org.
, 2.5.2 Inverse Cosine . . . . . . . 47
2.5.3 Inverse Tangent . . . . . . 48
2.5.4 Other Trigonometric Inverses 49
2.6 L’Hôpital’s Rule . . . . . . . . . . 54
2.6.1 Indeterminate Forms of
Contents type 0 · ∞ and ∞ − ∞ . .
2.6.2 Exponential Indeterminate
56
Forms . . . . . . . . . . . 58
Review Exercises . . . . . . . . . . . . . . 61
3 Integration Methods 63
1 Integration Review 1 3.1 Integration by Parts . . . . . . . . 64
1.1 The Meaning of the Definite Integral 2 3.2 Trigonometric Integrals . . . . . . 68
1.2 The Fundamental Theorem of 3.3 Trigonometric Substitution . . . . 74
Calculus . . . . . . . . . . . . . . 3 3.4 Partial Fraction Decomposition . 78
1.3 Indefinite Integrals . . . . . . . . 3 3.5 General Strategies for Integration 89
1.4 Integration by Substitution . . . . 4 3.6 Improper Integrals . . . . . . . . 92
1.5 Integration Examples . . . . . . . 4 3.6.1 Improper Integrals of the
Review Exercises . . . . . . . . . . . . . 7 First Kind . . . . . . . . . 92
3.6.2 Improper Integrals of the
2 Inverses and Other Functions 9 Second Kind . . . . . . . .
95
2.1 Inverse Functions . . . . . . . . . 10 Review Exercises . . . . . . . . . . . . .
100
2.1.1 Horizontal Line Test . . . 10
2.1.2 Finding Inverse Functions 13 4 Sequences and Series 101
2.1.3 Graphs of Inverse Functions 15 4.1 Sequences . . . . . . . . . . . . . 102
2.1.4 Derivative of an Inverse 4.2 Series . . . . . . . . . . . . . . . . . 111
Function . . . . . . . . . . 15 4.3 Testing Series with Positive Terms 118
2.1.5 Creating Invertible Functions 18 4.3.1 The Integral Test . . . . . 118
2.2 Exponential Functions . . . . . . 19 4.3.2 The Basic Comparison Test 123
2.2.1 The Natural Exponential 4.3.3 The Limit Comparison Test 125
Function . . . . . . . . . . 20 4.4 The Alternating Series Test . . . 129
2.2.2 Derivative of ex . . . . . . 22 4.5 Tests of Absolute Convergence . . 132
2.2.3 Integral of ex . . . . . . . 23 4.5.1 Absolute Convergence . . 132
2.2.4 Simplifying Exponential 4.5.2 The Ratio Test . . . . . . 134
Expressions . . . . . . . . 24 4.5.3 The Root Test . . . . . . . 135
2.3 Logarithmic Functions . . . . . . 26 4.5.4 Rearrangement of Series . 137
2.3.1 Logarithmic Function 4.6 Procedure for Testing Series . . . 138
Properties . . . . . . . . . 27 4.7 Power Series . . . . . . . . . . . . 143
2.3.2 The Natural Logarithmic
4.8 Representing Functions with
Function . . . . . . . . . . 28
Power Series . . . . . . . . . . . . . 151
2.3.3 Solving Exponential and
4.9 Maclaurin Series . . . . . . . . . . 157
Logarithmic Equations . . 29
4.10 Taylor Series . . . . . . . . . . . . . 161
2.3.4 Derivative of the Natural
Review Exercises . . . . . . . . . . . . . 166
Logarithmic Function . . . 32
2.3.5 Derivatives Using Answers 169
Arbitrary Bases . . . . . . 33
2.3.6 Logarithmic Differentiation 34 Summary 183
2.3.7 Integral of x1 and ax . . . 36
2.4 Exponential Growth and Decay . 40 Index 185
2.5 Inverse Trigonometric Functions . 45
2.5.1 Inverse Sine . . . . . . . . 45 GNU Free Documentation License 187
iii
, iv
