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When differentiating a function we find the derivative of the function. The theory of the derivatives and its applications in the investigation of the functions is covered in Differential Calculus. The fundamental problem of Integral Calculus is the inverse problem, i.e. given the derivative of a function to find the function. The solution of this inverse problem, (the integration of a given function), is of great importance in Mathematics, Physics and Engineering in general. However, this problem (integration) is more complicated as compared to the problem of differentiation. In very general terms we may say that integrals are classified as either Indefinite Integrals (functions) or as Definite Integrals (numbers). These two integrals are connected by the so called Fundamental Theorem of Calculus. In this first volume we cover the Indefinite Integrals. The Definite Integrals will be studied in details, in a second volume, to appear soon.
This book was written to provide an essential assistance to students who are first being introduced to the fundamentals of Integrals and has been designed to be an excellent supplementary textbook for University and College students in all areas of Mathematics, Physics and Engineering.
The content of the book is divided into 19 chapters, as shown analytically in the Table of Contents.
All fundamental techniques and methods of integration are presented in full details and with illustrative examples, (integration by parts, the substitution method, integration of rational functions of the integration variable, integration of functions which are rational with respect to the variable of integration x and the irrational functions of x entering into it, integration of the Binomial Differential, integration of trigonometric functions, integration of hyperbolic functions, integration with the aid of trigonometric andor hyperbolic substitutions, reduction or recurrence formulas etc.). Important applications of the Indefinite Integrals are considered in connection to the areas enclosed by curvilinear trapezoids and volumes of solids of revolution. Finally we consider some simple types of differential equations which are solved directly by means of appropriate integration techniques.
The text includes more than 120 illustrative worked out examples and 235 graded problems to be solved. The examples and the problems are designed to help the students to develop a solid background in the evaluation of Integrals, to broaden their knowledge and sharpen their analytical skills and finally to prepare them to pursue successful studies in more advanced courses in Mathematics. A brief hint or a detailed outline in solving more involved problems is often given. Finally answers to odd-numbered problems are also provided so that the students can check their progress and understanding of the material studied.
The Antiderivative Function
Some Simple Integration Techniques
Integration by Parts
The Substitution (Change of Variables) Method
Integration of Rational Functions
Evaluation of R
Evaluation of
Evaluation of
Integrals of Binomial Differentials
Integrating Trigonometric Functions
The Hyperbolic Functions
Integrating Hyperbolic Functions
Trigonometric and Hyperbolic Substitutions
Reduction Formulas
Miscellaneous Examples and Problems
Some General Remarks and Conclusions
Indefinite Integrals and Areas (An Introductory Approach)
Indefinite Integrals and Volumes of Revolution
Some Simple Types of Differential Equations