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

584 Pages·2019·18.46 MB·English
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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 NEW DELHI McGraw Hill Education Offices New Delhi New York St Louis San Francisco Auckland Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal San Juan Santiago Singapore Sydney Tokyo Toronto McGraw Hill Education (India) Private Limited Published by McGraw Hill Education (India) Private Limited, P-24, Green Park Extension, New Delhi – 110 016 Integral Calculus for JEE Main & Advanced, 2/e Copyright © 2013, by the McGraw Hill Education (India) Private Limited. No Part of this publication may be reproduced or distributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise or stored in a database or retrieval system without the prior written permission of the publishers. The program listings (if any) may be entered, stored and executed in a computer system, but they may not be reproduced for publication. This edition can be exported from India only by the publishers. McGraw Hill Education (India) Private Limited ISBN (13) : 978-1-25-906419-7 ISBN (10) : 1-25-906419-0 Vice President and Managing Director—McGraw Hill Education: India: Ajay Shukla Deputy General Manager—Test Prep and School: Tanmoy Roychowdhury Publishing Manager—Test Prep: K.N. Prakash Asst. Sponsoring Editor : Bhavna Malhotra Asst. Manager (Development Editing): Anubha Srivastava Asst. Manager—Production: Medha Arora Production Executive: Dharmender Sharma Product Specialist: Vikas Sharma General Manager—Production: Rajender P. Ghansela Production Manager: Reji Kumar Information contained in this work has been obtained McGraw Hill Education (India), from sources believed to be reliable. However, neither, McGraw Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw Hill Education (India) nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw Hill Education (India) and its authors are supplying information but are not attempting to render engineering or other professional services. If such ser- vices are required, the assistance of an appropriate professional should be sought. Typeset at Kaushik Laser Point & Printers, Tis Hazari Court, Delhi-53 and printed at Sanat Printers, 312 EPIP, HSIDC, Kundli, Sonepat, Haryana Cover printed at: Sanat Printers Cover Design: K. Anoop XXXXXXXXXXXXX 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

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