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Integration and Differential Equations R.S. Johnson Integration and differential equations Download free eBooks at bookboon.com 2 Integration and differential equations © 2012 R.S. Johnson & Ventus Publishing ApS ISBN 978-87-7681-970-5 Download free eBooks at bookboon.com 3 Integration and differential equations Contents Contents Preface to these two texts 8 Part I An introduction to the standard methods of elementary integration 9 List of Integrals 10 Preface 11 1 Introduction and Background 12 1.1 The Riemann integral 12 1.2 The fundamental theorem of calculus 17 1.4 Theorems on integration 19 1.5 Standard integrals 23 1.6 Integration by parts 26 1.7 Improper integrals 32 1.8 Non-uniqueness of representation 35 Exercises 1 36 2 The integration of rational functions 38 2.1 Improper fractions 38 2.2 Linear denominator, Q(x) 39 The next step for top-performing graduates Masters in Management Designed for high-achieving graduates across all disciplines, London Business School’s Masters in Management provides specific and tangible foundations for a successful career in business. This 12-month, full-time programme is a business qualification with impact. In 2010, our MiM employment rate was 95% within 3 months of graduation*; the majority of graduates choosing to work in consulting or financial services. As well as a renowned qualification from a world-class business school, you also gain access to the School’s network of more than 34,000 global alumni – a community that offers support and opportunities throughout your career. For more information visit www.london.edu/mm, email [email protected] or give us a call on +44 (0)20 7000 7573. * Figures taken from London Business School’s Masters in Management 2010 employment report Download free eBooks at bookboon.com 4 Click on the ad to read more Integration and differential equations Contents 2.3 Quadratic denominator, Q(x) 41 2.4 Cubic denominator, Q(x) 44 Exercises 2 48 3 The integration of trigonometric functions 49 3.1 Simple products 49 3.2 Powers of sin and cos 50 3.3 Rational functions of sin and cos 54 Exercises 3 56 Answers 58 Part II The integration of ordinary differential equations 60 List of Equations 61 Preface 62 1 What is a differential equation ? 63 1.1 The nature and solution of differential equations 63 1.2 Classification of ODEs 63 1.3 Overview of the equations to be discussed 65 CHALLENGING PERSPECTIVES Opportunities for Internships EADS unites a leading aircraft manufacturer, the world’s largest learning and development opportunities, and all the support you need, helicopter supplier, a global leader in space programmes and a you will tackle interesting challenges on state-of-the-art products. worldwide leader in global security solutions and systems to form Europe’s largest defence and aerospace group. More than We welcome more than 5,000 interns every year across 140,000 people work at Airbus, Astrium, Cassidian and Eurocopter, disciplines ranging from engineering, IT, procurement and in 90 locations globally, to deliver some of the industry’s most finance, to strategy, customer support, marketing and sales. exciting projects. Positions are available in France, Germany, Spain and the UK. An EADS internship offers the chance to use your theoretical knowledge and apply it first-hand to real situations and assignments To find out more and apply, visit www.jobs.eads.com. You can also during your studies. Given a high level of responsibility, plenty of find out more on our EADS Careers Facebook page. Download free eBooks at bookboon.com 5 Click on the ad to read more Integration and differential equations Contents 2 First order ODEs: standard results 67 2.1 Separable equations 67 2.2 Homogeneous equations 69 2.3 The general linear equation 73 3 First order ODEs: special equations 79 3.1 The Bernoulli equation 79 3.2 The Clairaut equation 80 3.3 The Riccati equation 84 3.4 Exact differentials 87 3.5 Missing variables 90 4 Second order ODEs 93 4.1 Constant coefficient equations 93 4.2 The Euler equation 96 4.3 Reduction of order 98 4.4 Variation of parameters 100 4.5 Finding particular integrals directly 102 4.6 Missing variables 105 Exercises 4 107 Excellent Economics and Business programmes at: “The perfect start of a successful, international career.” CLICK HERE to discover why both socially and academically the University of Groningen is one of the best www.rug.nl/feb/education places for a student to be Download free eBooks at bookboon.com 6 Click on the ad to read more Integration and differential equations Contents 5 More general aspects of ODEs 108 5.1 Interchanging x and y 108 5.2 Singular solutions 109 5.3 Uniqueness 112 Exercises 5 116 Answers 117 Index 119 Teach with the Best. Learn with the Best. Agilent offers a wide variety of affordable, industry-leading electronic test equipment as well as knowledge-rich, on-line resources —for professors and students. We have 100’s of comprehensive web-based teaching tools, lab experiments, application notes, brochures, DVDs/ See what Agilent can do for you. CDs, posters, and more. www.agilent.com/find/EDUstudents www.agilent.com/find/EDUeducators © Agilent Technologies, Inc. 2012 u.s. 1-800-829-4444 canada: 1-877-894-4414 Download free eBooks at bookboon.com 7 Click on the ad to read more Preface to these two texts These two texts in this one cover, entitled ‘An introduction to the standard methods of elementary integration’ (Part I) and ‘The integration of ordinary differential equations’ (Part II), are two of the ‘Notebook’ series available as additional and background reading to students at Newcastle University (UK). This pair constitutes a basic introduction to both the elementary methods of integration, and also the application of some of these techniques to the solution of standard ordinary differential equations. Thus the first is a natural precursor to the second, but each could be read independently. Each text is designed to be equivalent to a traditional text, or part of a text, which covers the relevant material, with many worked examples and some set exercises (for which the answers are provided). The necessary background is described in the preface to each Part, and there is a comprehensive index, covering the two parts, at the end. Download free eBooks at bookboon.com Part I An introduction to the standard methods of elementary integration Download free eBooks at bookboon.com Integration and differential equations List of Integrals List of Integrals List of Integrals This is a list of the integrals, and related problems, that constitute the worked examples considered here. This is a list of the integrals, and related problems, that constitute the worked examples considered here. 1 y= f() x = x2 (Riemann integral)…….…... p.14; e3x dx …..…… p.19; 0 − 3 primitives of 2x + 15 ,3− 4x3 2 , 3x + 4 1, 2sin 17x − 6, tan 1− 2x, , x2 +16 5 1+ 2x , x2sin x 3 , all p.25; 2 x2 −2 x − 8 x +x − 3 2 1 4 2 2 −2 2 π dx x+3 x − 1 d x , 3sec − xπ d x , ………all p.26; 2 2 1 0 1 9 − x 1 2 xn ln x d x ( x > 0 , n ≠ −1), arctan xd x , x2e−x d x , 4 − x2 d x ....all p.27; 0 0 2 1+ ln xn d x ( x > 0 , n ≠ −1), 1+ ln x3d x …..both p.30; I = tann θd θ ( n ≥ 2 ) 1 n ∞ ∞ ∞ dx dx tan6 θd θ …..both p.32; do , sin xd x , exist?…………….all p.33; 2 1+ x 0 1+ x 0 0 1 2 1 dx dx do , , ln xd x exist?………………all p.35; arctan(sinhx ) , 2arctan(ex ) , x 1+ x 0 −1 0 2arctan(tanh(x 2 )) , arcsin(tanhx ) all primitives of sechx d x …………..…..…p.36; x5−3 x 4 + 5 x 3 − 2 x 2 + x − 1 x3− x 2 +2 x + 1 divide ……….p.38; dx ……….....p.39; x2 +x +1 x +1 x6−3 x 5 + 2 x 4 + x 3 − x 2 +4 x − 7 x3 −1 dx ………p.40; dx ………..…..p.41; ()x − 2 3 x2 +4x + 3 2 x4−2 x 2 + 3 x − 2 x3 −3 x + 1 x6− x 3 +1 dx ....p.42; dx ….p.43; dx …..p.44; 2 2 3 2 1 x−6 x + 9 x −8x + 25 x +x +x +1 x − 2 x2 − x +1 dx …p.45; dx ..p.48; (x+1 )( x2 − 1 )( x 2 + 2 x + 2 )( x2 + 2 x + 5 ) (x−3 )( x2 + 4 x + 13 )2 π 2 sin2x cos 3 xd x ………….…..p.50; sin2x− sin 4 x sin 2 xd x …………...…...p.50; 0 π 2 sin10x cos 5 xd x ………..p.51; sin5x cos 2 xd x ……....p.52; sin4 xd x ……..p.53; 0 dx sin2 x …………..……………..p.54; dx ………..…………..p.55. 5+ 4cos x (2 + cosx )2 Download free eBooks at bookboon.com 10 Integration and differential equations List of Integrals Preface The material presented here is intended to provide an introduction to the methods for the integration of elementary functions. This topic is fundamental to many modules that contribute to a modern degree in mathematics and related studies – and especially those with a mathematical methods/applied/physics/engineering slant. The material has been written to provide a general introduction to the relevant ideas, and not as a text linked to a specific course of study that makes use of integration. Indeed, the intention is to present the material so that it can be used as an adjunct to a number of different modules or courses – or simply to help the reader gain a broader experience of, and an improved level of skill in, these important mathematical ideas.
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