Correct Antidifferentiation

A typical source of mistakes that frequently lead to a wrong or incomplete solution for the antiderivative of a given real function of one real variable is a misuse of the technique of change of variable. The increasing implementation of software in apparently mechanic tasks such as the calculation of antiderivatives has not improved the situation, yet those software packages issue generic warnings such as 'the answer's is not guaranteed to be continuous' or 'the solution might be only valid for parts of the function'. The practical meaning of those vague machine messages is clearly envisaged in this book, which shows how to handle the technique of change of variable in order to provide correct solutions.This book is monographically focused on elementary antidifferentiation and reasonably self-contained, yet it is written in a 'hand-book' style: it has plenty of examples and graphics in an increasing level of difficulty; the most standard changes of variable are studied and the hardest theoretic parts are included in a final Appendix. Each practical chapter has a list of exercises and solutions.This book is intended for instructors and university students of Mathematics of first and second year.Contents: Theoretical Chapters:Background (Sequences of Real Numbers and Compactness)Real-Valued Functions of a Real Variable (Continuity, Differentiability and Integrability in the Sense of Riemann)Elementary Real-Valued Functions (Power Functions, Root Functions, Polynomials, Exponential, Logarithm, Circular Functions and Hyperbolic Functions)Antidifferentiation (Fundamental Results; The Technique of Change of Variable and Their Tricks)Practical Chapters:Antidifferentiation by PartsRational FunctionsBinomial DifferentialsRational Functions of Trigonometric ArgumentsTablesand other papersAppendix
Readership: Instructors and university students of Mathematics of first and second year, University students of technical branches that include Calculus and/or Mathematical Analysis in their curriculum (Physics, Chemistry, Engineering, Business).Calculus;Mathematical Analysis;Real Functions;Real-Valued Functions of One Real Variable;Integral;Indefinite Integral;Antiderivative;Primitive;Integration;Antidifferentiation;Methods of Integration;Methods of Antidifferentiation;Change of Variable;Differentiability;Derivative;Riemann;Elementary Functions;Polynomials;Circular Functions;Hyperbolic Functions;Exponential;Logarithm;Methods of Integration;Methods of Antidifferentiation;Integration by Parts;Antidifferentiation by Parts;Quotient of Polynomials;Rational Functions;Hermite;Square Root of a Polynomial of the Second Degree;Binomial Differential;Beta Functions;Differential Binomial;Rational Function of Trigonometric Arguments;Integration Exercises;Antidifferentiation Exercises0Key Features:There is plenty of graphics which help the student to understand better the process of antidifferentiation while most of the books do not: a picture is worth a thousand words. Only the most standard changes of variable are studiedIt is written in a plain style. The hardest parts have been included in the final Appendix in order to prevent any detour from its main goal: calculation of antiderivativesThe book is reasonably self-contained for a student of first of second year of Math University studies. At this elementary level, it is possible to gather the results from other elaborated theories like Complex Analysis of Algebraic Geometry in a unified and understandable manner without recalling those mentioned theories which are usually taught in final courses