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 -