Engineering Mathematics I About the Author T Veerarajan retired as Dean, Department of Mathematics, Velammal College of Engineering and Technology, Viraganoor, Madurai, Tamil Nadu. A gold medalist from Madras University, he has had a brilliant academic career all through. He has 50 years of teaching experience at undergraduate and postgraduate levels in various established engineering colleges in Tamil Nadu including Anna University, Chennai. Engineering Mathematics I T Veerarajan Former Dean, Department of Mathematics Velammal College of Engineering and Technology Viraganoor, Madurai Tamil Nadu McGraw Hill Education (India) Private Limited Chennai McGraw Hill Education Offices Chennai 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 444/1, Sri Ekambara Naicker Industrial Estate, Alapakkam, Porur, Chennai 600 116 Engineering Mathematics I Copyright © 2018 by 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. 1 2 3 4 5 6 7 8 9 D102739 22 21 20 19 18 Printed and bound in India. ISBN (13): 978-93-87432-10-9 ISBN (10): 93-87432-10-6 Managing Director: Kaushik Bellani Director—Science & Engineering Portfolio: Vibha Mahajan Senior Portfolio Manager—Science & Engineering: Hemant K Jha Associate Portfolio Manager, Science & Engineering: Mohammad Salman Khurshid Production Head: Satinder S Baveja Assistant Manager—Production: Anuj K Shriwastava General Manager—Production: Rajender P Ghansela Manager—Production: Reji Kumar Information contained in this work has been obtained by McGraw Hill Education (India), from sources believed to be reliable. However, neither McGraw Hill Education (India) 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 services are required, the assistance of an appropriate professional should be sought. Typeset at Text-o-Graphics, B-1/56, Aravali Apartment, Sector-34, Noida 201 301, and printed at Cover Printer: Visit us at: www.mheducation.co.in Contents Preface ix Unit 1. Differential CalCUlUs 1.1–1.61 1.1 Functions 1.1 1.2 Limit of a Function 1.2 1.3 Continuity of a Function 1.5 1.4 Differentiability of a Function 1.5 Worked Example 1(a) 1.7 Exercise 1(a) 1.15 1.5 Derivatives 1.18 1.6 Rules of Differentiation 1.23 1.7 Derivatives of Hyperbolic Function 1.26 1.8 Methods of Differentiation 1.29 Worked Example 1(b) 1.30 Exercise 1(b) 1.44 1.9 Maxima and Minima of Functions of One Variable 1.46 Worked Example 1(c) 1.48 Exercise 1(c) 1.57 Answers 1.58 Unit 2. fUnCtions of several variables 2.1–2.69 2.1 Introduction 2.1 2.2 Total Differentiation 2.1 Worked Example 2(a) 2.4 Exercise 2(a) 2.20 2.3 Jacobians 2.27 2.4 Differentiation Under the Integral Sign 2.31 Worked Example 2(b) 2.33 Exercise 2(b) 2.48 2.5 Maxima and Minima of Functions of Two Variables 2.50 Worked Example 2(c) 2.52 Exercise 2(c) 2.65 Answers 2.67 vi Contents Unit 3. integral CalCUlUs 3.1–3.53 3.1 Introduction 3.1 3.2 Constant of Integration 3.1 3.3 Techniques of Integration 3.4 Worked Example 3(a) 3.6 Exercise 3(a) 3.13 3.4 Integration of Rational (Algebraic) Functions 3.14 Worked Example 3(b) 3.15 Exercise 3(b) 3.25 3.5 Integration of Irrational Functions 3.26 Worked Example 3(c) 3.27 Exercise 3(c) 3.35 3.6 Integration by Parts 3.36 Worked Example 3(d) 3.39 Exercise 3(d) 3.48 Answers 3.50 Unit 4. MUltiple integrals 4.1–4.83 4.1 Introduction 4.1 4.2 Evaluation of Double and Triple Integrals 4.1 4.3 Region of Integration 4.2 Worked Example 4(a) 4.3 Exercise 4(a) 4.15 4.4 Change of Order of Integration in a Double Integral 4.17 4.5 Plane Area as Double Integral 4.18 Worked Example 4(b) 4.21 Exercise 4(b) 4.38 4.6 Line Integral 4.41 4.7 Surface Integral 4.43 4.8 Volume Integral 4.44 Worked Example 4(c) 4.45 Exercise 4(c) 4.57 4.9 Gamma and Beta Functions 4.59 Worked Example 4(d) 4.63 Exercise 4(d) 4.77 Answers 4.79 Contents vii Unit 5. Differential eqUations 5.1–5.88 5.1 Equations of the First Order and Higher Degree 5.1 Worked Example 5(a) 5.3 Exercise 5(a) 5.12 5.2 Linear Differential Equations of Second and Higher Order with Constant Coefficients 5.14 5.3 Complementary Function 5.14 Worked Example 5(b) 5.19 Exercise 5(b) 5.32 5.4 Euler’s Homogeneous Linear Differential Equations 5.33 5.5 Simultaneous Differential Equations with Constant Coefficients 5.35 Worked Example 5(c) 5.35 Exercise 5(c) 5.47 5.6 Linear Equations of Second Order with Variable Coefficients 5.48 Worked Example 5(d) 5.53 Exercise 5(d) 5.67 5.7 Method of Variation of Parameters 5.71 Worked Example 5(e) 5.73 Exercise 5(e) 5.82 Answers 5.82 Preface I am deeply gratified with the enthusiastic response shown to all my books on Engineering Mathematics (for first year courses) by students and teachers throughout the country. This book has been designed to meet the requirements of the students of first-year undergraduate course on Engineering Mathematics. The contents have been covered in adequate depth for Semester I of various universities/deemed universities across the country. It offers a balanced coverage of both theory and problems. Lucid writing style supported with step-by-step solutions to all problems enhances understanding of the concepts. The book has ample number of solved and unsolved problems of different types to help students and teachers learning and teaching the subject. I hope that this book will be received by both the faculty and the students as enthusiastically as my other books on Engineering Mathematics. Critical evaluation and suggestions for the improvement of the book will be highly appreciated and acknowledged. T VEERARAJAN Publisher’s Note McGraw Hill Education (India) invites suggestions and comments from you, all of which can be sent to [email protected] (kindly mention the title and author name in the subject line). Piracy-related issues may also be reported.