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A Textbook ofENGINEERINGMATHEMATICS-I(SECOND EDITION)Prabhakar GuptaH.S. GangwarM.Sc. (Math.), M.Tech., Ph.D.Dean AcademicsSRMS College of Engineeringand Technology, Bareilly (U.P.)M.Sc., Ph.D.LecturerDeptt. of MathematicsSRMS College of Engineeringand Technology, Bareilly (U.P.)UPTUAn Imprint of

Copyright 2010, 2009 New Age International (P) Ltd., PublishersPublished by New Age International (P) Ltd., PublishersAll rights reserved.No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography,or any other means, or incorporated into any information retrieval system, electronic ormechanical, without the written permission of the publisher. All inquiries should beemailed to [email protected] (13) : 978-81-224-2847-6PUBLISHING FOR ONE WORLDNEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS4835/24, Ansari Road, Daryaganj, New Delhi - 110002Visit us at

Preface to theSecond Revised EditionThis book has been revised exhaustively according to the global demands of the students.Attention has been taken to add minor steps between two unmanageable lines where essential sothat the students can understand the subject matter without mental tire.A number of questions have been added in this edition besides theoretical portion wherevernecessary in the book. Latest question papers are fully solved and added in their respective units.Literal errors have also been rectified which have been accounted and have come to ourobservation. Ultimately the book is a gift to the students which is now error free and user- friendly.Constructive suggestions, criticisms from the students and the teachers are always welcomefor the improvement of this book.AUTHORS

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Some Useful Formulae1.2.sin ix i sin hx3.sin x e ix e ix2i4.cos x e ix e ix25.Sin h2x 1(cosh 2x – 1)26.cos h2x 1(cosh 2x 1) x dx axa 1, a 0log azzzsin haxdx 1cos haxacos haxdx 1sin haxatan haxdx 1log cos haxa8. ix cos hxzzzzzzz1a2 x212x a12x a22dx sin 1xx arc sinaadx log x x 2 a 2dx 1xxtan 1 arc tanaaaa 2 x 2 dx x2a2 x2 a2xsin 1a2e ax sin bx dx e ax(a sin bx – b cos bx)a2 b2e ax cos bx dx e ax(a cos bx b sin bx)a 2 b2sec ax dx 1log sec ax tan ax a

1log cosec ax – cot ax a18.z19.sin x x x 3 x5 .3520.cos x 1 x2 x 4 .2421.tan x x x32 5 17 7 x x .3 1531522.log (1 x) x 23.log (1 – x) x 24.sin hx x 25.d xa ax loge adx26.1dcos 1 x dx1 x227.1dcot 1 x dx1 x228.29.30.cosec ax dx ddxddxx2 x3 .23x 3 x5 .35cosec 1 x log a x x2 x3 x 4 .2341x1x x2 1log a edxatan 1. 2dxaa x2

ContentsPREFACE TO THE SECOND REVISED EDITIONSOME USEFUL FORMULAEU NIT I. Differential Calculus-I1.0 Introduction1.1 nth Derivative of Some Elementary FunctionsExercise 1.11.2 Leibnitz’s TheoremExercise 1.2Exercise 1.3Partial Differentiation1.3 Function of Two Variables1.4 Partial Differential CoefficientsExercise 1.41.5 Homogeneous Function1.6 Euler’s Theorem on Homogeneous FunctionsExercise 1.51.7 Total Differential CoefficientExercise 1.6Curve Tracing1.8 Procedure for Tracing Curves in Cartesian FormExercise 1.71.9 Polar CurvesExercise 1.81.10 Parametric CurvesExercise 1.9Expansion of Function of Several Variables1.11 Taylor’s Theorem for Functions of Two VariablesExercise 1.10Objective Type QuestionsAnswers to Objective Type QuestionsU NIT II. Differential Calculus-II2.1 JacobianExercise 2.11 9495 15095109

2.2 Approximation of ErrorsExercise 2.22.3 Extrema of Function of Several VariablesExercise 2.32.4 Lagrange’s Method of Undetermined MultipliersExercise 2.4Objective Type QuestionsAnswers to Objective Type QuestionsU NIT III. tion of MatrixTypes of MatricesOperations on MatricesTrace of MatrixProperties of TransposeProperties of Conjugate MatricesSingular and Non-Singular MatricesAdjoint of a Square MatrixInverse of a Matrix (Reciprocal)Exercise 3.1Elementary Row and Column TransformationsMethod of Finding Inverse of a Non-Singular Matrix by ElementaryTransformationsExercise 3.2Rank of a MatrixExercise 3.3System of Linear Equations (Non-Homogeneous)System of Homogeneous EquationsGaussian Elimination MethodExercise 3.4Linear Dependence of VectorsExercise 3.5Eigen Values and Eigen VectorsExercise 3.6Cayley-Hamilton TheoremExercise 3.7Diagonalization of a MatrixApplication of Matrices to Engineering ProblemsExercise 3.8Objective Type QuestionsAnswers to Objective Type Questions111119121134135145147150151 6188197200206210214214230232238239249253255257

U NIT IV. Multiple IntegralsDouble IntegralsWorking RuleDouble Integration for Polar CurvesExercise 4.1Change of the Order of IntegrationChange of Variables in a Multiple IntegralExercise 4.2Beta and Gamma FunctionsTransformations of Gamma FunctionTransformations of Beta FunctionRelation between Beta and Gamma FunctionsSome Important DeductionsDuplication FormulaEvaluate the IntegralsExercise 4.3Application to Area (Double Integrals)Exercise 4.4Triple IntegralsExercise 4.5Application to Volume (Triple Integrals)Exercise 4.6Dritchlet’s TheoremExercise 4.7Objective Type QuestionsAnswers to Objective Type QuestionsU NIT V. Vector CalculusVector Differential Calculus5.1 Vector Function5.2 Vector Differentiation5.3 Some Results on DifferentiationExercise 5.15.4 Scalar Point Function5.5 Vector Point Function5.6 Gradient or Slope of Scalar Point Function5.7 Geometrical Meaning of Gradient, Normal5.8 Directional Derivative5.9 Properties of GradientExercise 5.25.10 Divergence of a Vector Point Function5.11 Physical Interpretation of Divergence5.12 Curl of a Vector258 9300311312314315322323329330332333 418333333333334336337337337338338339350351352353

5.13 Physical Meaning of Curl5.14 Vector IdentitiesExercise 5.35.15 Vector Integration5.16 Line Integral5.17 Surface Integral5.18 Volume IntegralExercise 5.45.19 Green’s TheoremExercise 5.55.20 Stoke’s Theorem5.21 Cartesian Representation of Stoke’s TheoremExercise 5.65.22 Gauss’s Divergence Theorem5.23 Cartesian Representation of Gauss’s TheoremExercise 5.7Objective Type QuestionsAnswers to Objective Type 0401413414418Unsolved Question Papers (2004 2009)419 431Index433 434

UNIT1Differential Calculus-I1.0INTRODUCTIONCalculus is one of the most beautiful intellectual achievements of human being. The mathematicalstudy of change motion, growth or decay is calculus. One of the most important idea of differentialcalculus is derivative which measures the rate of change of a given function. Concept of derivativeis very useful in engineering, science, economics, medicine and computer science.dyd2 y, second order derivative, denoted bydxdx 2d3 ythird order derivative byand so on. Thus by differentiating a function y f(x), n times,dx 3dnyor Dny or yn(x). Thus, the processsuccessively, we get the nth order derivative of y denoted bydx nof finding the differential co-efficient of a function again and again is called SuccessiveDifferentiation.The first order derivative of y denoted by1.1 nth DERIVATIVE OF SOME ELEMENTARY FUNCTIONS1. Power Function (ax b)mLety (ax b)my1 ma (ax b)m–1y2 m (m–1)a2 (ax b)m–2. . . . .yn m(m–1) (m–2) . (m – n – 1) an (ax b)m–nCase I. When m is positive integer, thenm (m – 1).(m – n 1)(m – n).3 2 1 nyn a (ax b)m–n(m – n).3 2 1yn mdnm( ) axba n ( ax b) m nm ndx n1

2A TEXTBOOK OF ENGINEERING MATHEMATICS—ICase II. When m n ve integernyn 0a n (ax b)0 n an dn( ax b)n n a nndxCase III. When m –1, theny (ax b)–1 1( ax b)yn (–1) (–2) (–3) . (–n) an (ax b)–1–n RSTdn1ndx ax bUVW( 1)n n an (ax b)n 1Case IV. Logarithm case: When y log (ax b), thenaax bDifferentiating (n–1) times, we gety1 yn and n 1( ax b) 1dx n 1Using case III, we obtain lqdnlog( ax b)dx n ( 1) n 1 ( n 1) a n( ax b) n2. Exponential Function(i) Considery amxy1 mamx. loge ay2 m2amx (loge a)2.yn mn amx (loge a)n(ii) ConsiderPuttingy emxa e in aboveyn mnemx3. Trigonometric Functions cos (ax b) or sin (ax b)Lety cos (ax b), thenF ax b π IH2K2π IFcos ax b H2 K3π IFcos ax b H2 Ky1 – a sin (ax b) a cosy2 – a2 cos (ax b) a2y3 a3 sin (ax b) a3.

3DIFFERENTIAL CALCULUS-Iyn Similarly,yn FIHKnπ IFsin ax b H2Knπdncos ( ax b) a n cos ax b n2dxdndxnsin( ax b) a n4. Product Functions eax sin (bx c ) or e ax cos (bx c )Consider the function y eax sin (bx c)y1 eax·b cos (bx c) aeax sin (bx c) eax [b cos (bx c) a sin (bx c)]To rewrite this in the form of sin, puta r cos φ, b r sin φ, we gety1 eax [r sin φ cos (bx c) r cos φ sin (bx c)]y1 reax sin (bx c φ)Here,r –1a 2 b 2 and φ tan (b/a)Differentiating again w.r.t. x, we gety2 raeax sin (bx c φ) rbeax cos (bx c φ)Substituting for a and b, we gety2 reax. r cos φ sin (bx c φ) reax r sin φ cos (bx c φ)y2 r2eax [cos φ sin (bx c φ) sin φ cos (bx c φ)] r2 eax sin (bx c φ φ)y2 r2 eax sin (bx c 2φ)Similarly,y3 r3eax sin (bx c 3φ).yn dndxne ax sin( bx c ) r n e ax sin (bx c nφ)In similar way, we obtaind n axe cos(bx c) r ne ax cos (bx c nφ)yn dx n11 5x 6x 21 ( 2 x 1)( 3 x 1)Example 1. Find the nth derivative ofSol. Letor 11 5x 6x 223 y 2x 1 3x 1y yn 2(By Partial fraction)dndn–1(2x–1)–3(3x – 1)–1dx ndx n

4A TEXTBOOK OF ENGINEERING MATHEMATICS—IL( 1) n 2 OP LM( 1) n 3 OP2 MMN (2x 1) PQ – 3 MN (3x 1) PQL 2 3 OP .( 1) n MN (2x 1) ( 3x 1) Qn oryn nnn 1n 1n 1Example 2. Find the nth derivative of eax cos2 x sin x.(1 cos 2 x)sin xSol. Lety eax cos2 x sin x eax21 ax1 axe sin x e ( 2 cos 2x sin x) 22 21 ax1e sin x e ax sin( 3x) sin x 241 ax1 axe sin x e sin 3xy 441 n ax1 yn r e sin (x nφ) r1 n e ax sin ( 3x nθ) .44whereand( 1) n n a ndn 1(ax b) dx n( ax b) n 1n 1lorAsn 1n 1nnr qa 2 1 ; tan φ 1/aa 2 9 ; tan θ 3/a.2x, find yn.Example 3. If y tan–11 x22xSol. We havey tan 11 x2Differentiating y w.r.t. x, we getr1 y1 11 F 2x IH1 x K2 (U.P.T.U., 2002)FGHd2xdx 1 x 222(1 x 2 ) 2 IJK e1 x(1 x 2 ) 24 2x 2 4x 2(1 x 2 ) 2y1 1 11 , (by Partial fractions)i x i x iLMNOPQDifferentiating both sides (n–1) times w.r. to ‘x’, we getyn LMMNn 1n 11 ( 1) ( n 1) ( 1) ( n 1) i( x i) n( x i) n( 1)n 1 ( n 1)i 2(1 x 2 ) 4x 2(1 x 2 ) 22( x i)( x i )y1 (1 x 2 )j ( x i ) n ( x i ) n( 1)n 1 ( n 1)iOPPQr n (cos θ i sin θ) n r n (cos θ i sin θ) n(where x r cos θ, 1 r sin θ)

5DIFFERENTIAL CALCULUS-I ( 1) n 1 ( n 1) r nin 1 r–n sin nθ, where r yn 2(–1)n–1x 1. Show thatx 1LM x nN (x 1)n 2yn (–1)n–2Sol. We havex2 1F 1I.H xKθ tan–1Example 4. If y x logcos nθ i sin nθ cos nθ i sin nθn x n( x 1)nOPQ(U.P.T.U., 2002)x 1 x [log(x – 1) – log (x 1)]x 1y x logDifferentiating w.r. to ‘x’, we getLM 1 1 OPN x 1 x 1QF 1 I F 1 1 I log (x – 1) – log (x 1) 1 H x 1K H x 1 Ky1 log (x – 1) – log (x 1) xory1 log (x –1) – log (x 1) 11 x 1 x 1Differentiating (n–1) times with respect to N, we getyn d n 1dx n 1log (x 1) d n 2d n 1log (x 1) dx n 1d n 1d n 1dxdx n 1(x 1) 1 n 1RS d log(x 1)UV d RS d log(x 1)UV ( 1) n 1 ( 1) n 1TdxW dx TdxW (x 1)(x 1)n 2dxn 2(x 1) 1n 1n 2FHIKFHn 1nnIK( 1)n 1 n 1 ( 1) n 1 n 1d n 2d n 211 (x 1) n( x 1) ndx n 2 x 1dx n 2 x 1( 1)n 2 n 2(x 1) n 1 ( 1) n 2 n 2(x 1)n 1LM x 1N (x 1)L x nn 2MN (x 1)n 2n 2 ( 1)n n 2 ( 1)n Example 5. Find yn (0) if y Sol. We havey x 1( x 1)nx n( x 1)x3x 12n( 1)n 1 (n 1) n 2 OP .Q(x 1)n( n 1)( x 1)n ( n 1)( x 1)n ( 1)n 1 (n 1) n 2OPQ.x3x 3 1 1 ( x 1)( x 2 x 1)1 2 22( x 1)( x 1)x 1x 1x 1(x 1)n

6A TEXTBOOK OF ENGINEERING MATHEMATICS—Iory x2 x 11 ( x 1)( x 1)( x 1)y x2 1 11 1 x 1x 1 x 1y ory a fa f11L 11 O M x x 1 2 N x 1 x 1 PQ1L 11 O x M2 N x 1 x 1 PQ( 1) n O1 L ( 1) nP 0 M2 NM ( x 1)( x 1) QP( 1) n L11MN (x 1) (x 1) OPQ2( 1) n L 11 O M2N ( 1) (1) PQn yn nn 1n 1noryn n 1n 1nAtx 0, yn (0) When n is odd, yn(0) When n is even, yn(0) n 1( 1) n n2( 1) n n2n 11 1 n 1 1 0 .EXERCISE 1.11 . If y x2, find nth derivative of y.( x 1)2 ( x 2)(U.P.T.U., 2002)LMAns. yMN2 . Find the nth derivative ofx2.( x a)( x b)3 . Find the nth derivative of tan–1LM 1 x OP .N1 x Qn ( 1) n n 13( x 1) n 2LMAns.MN ( 1) n n( a b)5( 1) n n9( x 1) n 1LM aN ( x a)2n 1 OPPQOPOPQPQ4( 1) n n9( x 2) n 1b2(x b)n 1[Ans. ( 1)n 1 n 1 sin n θ sin nθ where θ cot–1x ]4 . If y sin3 x, find yn.5 . Find nth derivative of tan–1F xI .H aKLMAns.NF IH KAns. a 1fn 1 aFHππ31 . 3n . sin 3x nsin x n4242n 1 nI OPKQsin n θ sin nθ

7DIFFERENTIAL CALCULUS-Ia f2( 1) n OPP( x 1)QAns. e x x n6 . Find yn, where y ex.x.LMAns.MNAns. a 1

ee i ix ix 2 4.4.4. cos x . PREFACE TO THE SECOND REVISED EDITION SOME USEFUL FORMULAE 1.0 Introduction 1 1.1 nth Derivative of Some Elementary Functions 1 Exercise 1.1 6 1.2 Leibnitz’s Theorem 7 Exercise 1.2 13 Exercise 1.3 19 Partial Differentiation 20 1.3 Function of Two Variables 20 1.4 Partial Differential Coefficients 21 Exercise 1.4 33 1.5 Homogeneous Function 35 1.6 .