Bibliography textbooks

1. Bermant A.F., Aramanovich. Mathematical analysis. A brief course for engineering students. – Moscow: Mir Publishers, 1975. - 782 p.

2. Howard Anton. Calculus. Third edition.

3. Krasnov M., Kiselev A., Makarenko G., Shikin E. Mathematical analysis for engineers.

4. Piscunov N. Differential and integral calculus. – Moscow: Mir Publishers, 1969. - 895 p.

5. Yakovlev G.N. Higher mathematics. – Moscow: Mir Publishers, 1990, - 480 p.

6. Кремер Н.Ш. и др. Высшая математика для экономистов. Уч. Пособие для вузов.- М.: Банки и биржи, ЮНИТИ, 1997. - 439 с.

7. Кремер Н.Ш. Математика в экономике. М.: Финстатинформ, 1999.

8. Общий курс высшей математики для экономистов: Учебник/Под ред. В.И.Ермакова. - М.: ИНФРА, 1999. - 656 с.

9. Пак В.В., Носенко Ю.Л. Вища математика: Підручник. – Д.: „Видав-ництво Сталкер”, 2003. – 496 с.

10. Пискунов Н.С. Дифференциальное и интегральное исчисления. Т.1, 2. – М.: Наука, 1978, 456, 576 с..

11. Шнейдер В.Е., Слуцкий А.И., Шумов А.С. Краткий курс высшей математики. Тт. 1, 2. Учеб. Пособие для втузов. – М.: „Высшая школа”, 1978. – 384, 328 с.

Problem books

1. Berman G. Problem book on mathematical analysis

2. Берман Г.Н. Сборник задач по курсу математического анализа

3. Руководство к решению задач с экономическим содержанием по курсу высшей математики. Под ред. А.И. Карасева и Н.Ш.Кремера. М.: Экономическое образование, 1989.

4. Сборник задач по высшей математике для экономистов: Учебное пособие / Под ред. В.И. Ермакова. – М.: ИНФРА-М, 2002. – 575 с.

Contents

INTEGRAL CALCULUS 5

LECTURE NO. 19. PRIMITIVE AND INDEFINITE INTEGRAL 5

POINT 1. PRIMITIVE 5

POINT 2. INDEFINITE INTEGRAL AND ITS PROPERTIES 6

POINT 3. INTEGRATION BY SUBSTITUTION (CHANGE OF A VARIABLE) 9

POINT 4. INTEGRATION BY PARTS 13

LECTURE NO.20. CLASSES OF INTEGRABLE FUNCTIONS 17

POINT 1. RATIONAL FUNCTIONS (RATIONAL FRACTIONS) 17

POINT 2. TRIGONOMETRIC FUNCTIONS 20

POINT 3. IRRATIONAL FUNCTIONS 26

INDEFINITE INTEGRAL: Basic Terminology 30

LECTURE NO. 21. DEFINITE INTEGRAL 35

POINT 1. PROBLEMS LEADING TO THE CONCEPT OFA DEFINITE INTEGRAL 35

POINT 2. DEFINITE INTEGRAL 37

POINT 3. PROPERTIES OF A DEFINITE INTEGRAL 39

POINT 4. DEFINITE INTEGRAL AS A FUNCTION OF ITS UPPER VARIABLE LIMIT 42

POINT 5. NEWTON-LEIBNIZ FORMULA 45

POINT 6. MAIN METHODS OF EVALUATION A DEFINITE INTEGRAL 47

Change of a variable (substitution method) 47

Integration by parts 48

LECTURE NO.22. APPLICATIONS OF DEFINITE INTEGRAL 51

POINT 1. PROBLEM – SOLVING SCHEMES. AREAS 51

POINT 2. ARС LENGTH 56

POINT 3. VOLUMES 59

LECTURE NO. 23. DEFINITE INTEGRAL: ADDITIONAL QUESTIONS 64

POINT 1. APPROXIMATE INTEGRATION 64

Rectangular Formulas 64

Trapezium Formula 65

Simpson formula (parabolic formula) 66

POINT 2. IMPROPER INTEGRALS 69

Improper integrals of the first kind 69

Improper integrals of the second kind 72

Convergence tests 75

POINT 3. EULER Г- FUNCTION 77

DEFINITE INTEGRAL: Basic Terminology 79

LECTURE NO. 24. DOUBLE INTEGRAL 85

POINT 1. DOUBLE INTEGRAL 85

POINT 2. EVALUATION OF A DOUBLE INTEGRAL IN CARTESIAN COORDINATES 87

POINT 3. IMPROPER DOUBLE INTEGRALS. POISSON FORMULA 93

POINT 4. DOUBLE INTEGRAL IN POLAR COORDINATES 94

DOUBLE INTEGRAL: Basic Terminology 98

DIFFERENTIAL EQUATIONS 100

LECTURE NO.25. FIRST AND SECOND ORDER DIFFERENTIAL EQUATIONS 100

POINT 1. GENERAL NOTIONS 100

POINT 2. INTEGRABLE TYPES OF THE FIRST ORDER DIFFERENTIAL EQUATIONS (of DE - 1) 103

1. Separated DE-1 (DE-1 with separated variables) 103

2. Separable DE-1 (DE-1 with separable variables) 105

3. Homogeneous DE-1 109

4. Linear DE-1 113

5. Bernoulli DE-1 116

POINT 3. ORDER REDUCING SECOND ORDER DIFFERENTIAL EQUATIONS 117

LECTURE NO.26. SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS 123

POINT 1. GENERAL NOTIONS 123

POINT 2. LINEAR DEPENDENCE AND INDEPENDENCE 124

POINT 3. HOMOGENEOUS EQUATIONS 128

Structure of the general solution of SO LHDE 128

SO LHDE with constant coefficients 129

POINT 4. NONHOMOGENEOUS EQUATIONS 131

Structure of the general solution of SO LNDE 131

Method of variation of arbitrary constants 132

Method of undetermined coefficients for SO LNDE with constant coefficients 135

LECTURE NO. 27. SYSTEMS OF DIFFERENTIAL EQUATIONS. APPROXIMATE INTEGRATION OF DIFFERENTIAL EQUATIONS 142

POINT 1. NORMAL SYSTEMS OF DIFFERENTIAL EQUATIONS 142

POINT 2. APPROXIMATE INTEGRATION OF DIFFERENTIAL EQUATIONS 145

Successive approximations method 145

Euler method 147

DIFFERENTIAL EQUATIONS: Basic Terminology 150

BIBLIOGRAPHY 158

CONTENTS 159

Integral calculus. Differential equations (Інтеґральне числення. Диференціальні рівняння): Методичний посібник по вивченню розділів курсу ”Математичний аналіз” для студентів ДонНТУ (англійською мовою)

УКЛАДАЧ: Косолапов Юрій Федорович, кандидат фізико-математич-них наук, професор

ФОРМАТ 60×84 . Умовних друкарських аркушів

83000, м. Донецьк, вул. Артема, 58, ДонНТУ

1 Cauchy, A.L. (1780 - 1859), an eminent French mathematician

2 Riemann G.F.B. (1826 - 1866), an eminent German mathematician

3 Bolzano, B. (1781 - 1848), a Czech mathematician, philosopher, and logician

4 Newton, I. (1642 - 1727), the great English scientist

5 Leibniz, G. (1646 – 1717), the great German philosopher and mathematician

6 See Lect. No. 8, Point 4, Ex. 12

7 See Lecture NO. 4, Point 2, Ex. 4

8 Ib., Ex. 5

9 That is or .

10 Simpson, T. (1710 - 1761), an English mathematician

11 Agnesi, M.G. (1718 - 1799), an Italian mathematician