Integral Calculus for IIT JEE Main and Advanced Vinay Kumar VKR Classes Kota

584 Pages·2019·18.46 MB·English

by  Vinay Kumar VKR Classes Kota

for
A A
BOUT THE UTHOR
Vinay Kumar (VKR) graduated from IIT Delhi in
Mechanical Engineering.
Presently, he trains IIT aspirants at VKR Classes,
Kota, Rajasthan.
for
Second Edition
Vinay Kumar
B.Tech., IIT Delhi
McGraw Hill Education (India) Private Limited
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Integral Calculus for JEE Main & Advanced, 2/e
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PREFACE
This book is meant for students who aspire to join the Indian Institute of Technologies (IITs) and
various other engineering institutes through the JEE Main and Advanced examinations. The content
has been devised to cover the syllabi of JEE and other engineering entrance examinations on the topic
Integral Calculus. The book will serve as a text book as well as practice problem book for these competi-
tive examinations.
As a tutor with more than thirteen years of teaching this topic in the coaching institutes of Kota,
I have realised the need for a comprehensive textbook in this subject.
I am grateful to McGraw-Hill Education for providing me an opportunity to translate my years of
teaching experience into a comprehensive textbook on this subject.
This book will help to develop a deep understanding of Integral Calculus through concise theory
and problem solving. The detailed table of contents will enable teachers and students to easily access
their topics of interest.
Each chapter is divided into several segments. Each segment contains theory with illustrative ex-
amples. It is followed by Concept Problems and Practice Problems, which will help students assess the
basic concepts. At the end of the theory portion, a collection of Target Problems have been given to
develop mastery over the chapter.
The problems for JEE Advanced have been clearly indicated in each chapter.
The collection of objective type questions will help in a thorough revision of the chapter. The
Review Exercises contain problems of a moderate level while the Target Exercises will assess the students’
ability to solve tougher problems. For teachers, this book could be quite helpful as it provides numerous
problems graded by difficulty level which can be given to students as assignments.
I am thankful to all teachers who have motivated me and have given their valuable recommenda-
tions. I thank my family for their whole-hearted support in writing this book. I specially thank Mr. Devendra
Kumar and Mr. S. Suman for their co-operation in bringing this book.
Suggestions for improvement are always welcomed and shall be gratefully acknowledged.
Vinay Kumar

CONTENT
About the Author ii
Preface v
CHAPTER 1 INDEFINITE INTEGRATION 1.1 – 1.184
1.1 Introduction 1.1
1.2 Elementary Integrals 1.4
1.3 Integration by Transformation 1.10
1.4 Integration by Substitution 1.16
1.5 Integrals Involving Sine and Cosine 1.27
1.6 Rationalization by Trigonometric Substitution 1.36
1.7 Integrals of the Form 1.40
1.8 Integrals of the Form 1.45
1.9 Integrals of the Form 1.50
1.10 Integration of Trigonometric Functions 1.55
1.11 Integration by Parts 1.65
1.12 Special Integrals 1.73
1.13 Multiple Integration by Parts 1.76
1.14 Integration by Reduction Formulae 1.81
1.15 Integration of Rational Functions Using Partial Fractions 1.88
1.16 Special Methods For Integration of Rational Functions 1.101
1.17 Integration of Irrational Functions 1.106
dx
1.18 Integrals of the Type 1.112
P Q
1.19 Integration of a Binomial Differential 1.118
1.20 Euler’s Substitution 1.120
1.21 Method of Undetermined Coefficients 1.124
viii | Content
1.22 Non-elementary Integrals 1.127
Target Problems for JEE Advanced 1.130
Things to Remember 1.142
Objective Exercises 1.147
Review Exercises for JEE Advanced 1.160
Target Exercises for JEE Advanced 1.162
Previous Year’s Questions (JEE Advanced) 1.164
Answers 1.166
CHAPTER 2 DEFINITE INTEGRATION 2.1 – 2.204
2.1 Introduction 2.1
2.2 Definite Integral as a Limit of Sum 2.5
2.3 Rules of Definite Integration 2.12
2.4 First Fundamental Theorem of Calculus 2.19
2.5 Second Fundamental Theorem of Calculus 2.27
2.6 Integrability 2.41
2.7 Improper Integral 2.52
2.8 Substitution in Definite Integrals 2.63
2.9 Integration by parts for Definite Integrals 2.72
2.10 Reduction Formula 2.78
2.11 Evaluation of Limit of sum using Newton-leibnitz Formula 2.83
2.12 Leibnitz Rule for Differentiation of Integrals 2.91
2.13 Properties of Definite Integral 2.95
2.14 Additional Properties 2.122
2.15 Estimation of Definite Integrals 2.124
2.16 Determination of Function 2.134
2.17 Wallis’ Formula 2.138
2.18 Limit under the sign of Integral 2.143
2.19 Differentiation under the sign of Integral 2.144
2.20 Integration of Infinite Series 2.148
2.21 Approximation of Definite Integrals 2.151
Target Problems for JEE Advanced 2.154
Things to Remember 2.166
Objective Exercises 2.169
Review Exercises for JEE Advanced 2.180
Target Exercises for JEE Advanced 2.184
Content | ix
Previous Year’s Questions (JEE Advanced) 2.188
Answers 2.194
CHAPTER 3 AREA UNDER THE CURVE 3.1 – 3.86
3.1 Curve Sketching 3.1
3.2 Area of a Curvilinear Trapezoid 3.7
3.3 Area Bounded by a Function which Changes Sign 3.10
3.4 Area of a Region Between two Non-intersecting Graphs 3.13
3.5 Area of a Region Between two intersecting Graphs 3.17
3.6 Area by Horizontal Strips 3.21
3.7 Area of a Region Between Several Graphs 3.26
3.8 Determination of Parameters 3.30
3.9 Shifting of Origin 3.35
3.10 Area Bounded by a Closed Curve 3.37
3.11 Areas of Curves given by Parametric Equations 3.42
3.12 Areas of Curves given by Polar Equations 3.44
3.13 Areas of Regions given by Inequalities 3.46
Target Problems for JEE Advanced 3.51
Things to Remember 3.62
Objective Exercises 3.64
Review Exercises for JEE Advanced 3.74
Target Exercises for JEE Advanced 3.76
Previous Year’s Questions (JEE Advanced) 3.78
Answers 3.80
CHAPTER 4 DIFFERENTIAL EQUATIONS 4.1 – 4.99
4.1 Introduction 4.1
4.2 Formation of a Differential Equation 4.3
4.3 Solution of a Differential Equation 4.7
4.4 First Order and First Degree Differential Equations 4.11
4.5 Reducible to Variable Separable 4.17
4.6 Homogeneous Differential Equations 4.23
4.7 Linear Differential Equations 4.30
4.8 Solution by Inspection 4.38
4.9 First Order Higher Degree Differential Equation 4.42