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VŠB –TECHNICAL UNIVERSITY OF OSTRAVA Department of Mathematics and Descriptive Geometry MATHEMATICS II Pavel Kreml Ostrava CONTENTS 1. INDEFINITE INTEGRALS ................................................................................. 3 1.1. The Indefinite Integral .............................................................................................................................. 3 1.2. Computation of Integrals ......................................................................................................................... 3 1.3. Some Properties of the Indefinite Integral .............................................................................................. 5 1.4. Substitution ................................................................................................................................................ 5 1.4. Integration by Parts .................................................................................................................................. 7 2. DEFINITE INTEGRALS ..................................................................................... 9 2.1. The Definite Integral ................................................................................................................................. 9 2.2. The Area Problem and the Definite Integral ........................................................................................ 10 2.3. Rules for Definite Integrals .................................................................................................................... 11 2.4. Integration by Parts in Definite Integrals ............................................................................................. 12 2.5. Substitution in Definite Integrals ........................................................................................................... 12 2.6. Applications of Integration ..................................................................................................................... 13 Finding Areas under Curves .................................................................................................................. 13 Area Between Two Curves .................................................................................................................... 14 Volumes of Revolution .......................................................................................................................... 15 The Length of a Curve ........................................................................................................................... 16 Area of Surface of Revolution ............................................................................................................... 17 3. DIFFERENTIAL EQUATIONS ........................................................................ 18 3.1. Introduction ............................................................................................................................................. 18 3.2. Separable Equations ............................................................................................................................... 20 3.3. Linear Differential Equations ................................................................................................................ 21 Homogeneous Linear Differential Equation .......................................................................................... 21 Nonhomogeneous Linear Differential Equation .................................................................................... 23 Method for Solving Linear Differential Equation .................................................................................. 23 th 3.4. Linear Differential Equations n order ................................................................................................ 25 3.5. Homogeneous Equations with Constant Coefficients .......................................................................... 26 3.6. Homogeneous 2nd Order Equations with Constant Coefficients ......................................................... 27 I. Real Distinct Roots: ............................................................................................................................ 27 II. Repeated Roots: ................................................................................................................................ 27 III. Complex Conjugate Roots: .............................................................................................................. 28 3.6. Nonhomogeneous Equations with Constant Coefficients .................................................................... 29 Method of Variation of Parameters ........................................................................................................ 30 Method of Undetermined Coefficients .................................................................................................. 32 LITERATURE ........................................................................................................... 35 - 1 -
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