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Problem Books in Mathematics Series Editor: P.R. Halmos Unsolved Problems in Intuitive Mathematics, Volume I: Unsolved Problems in Number Theory by Richard K. Guy 1981. xviii, 161 pages. 17 illus. Theorems and Problems in Functional Analysis by A.A. Kirillov and A.D. Gvishiani 1982. ix, 347 pages. 6 illus. Problems in Analysis by Bernard Gelbaum 1982. vii, 228 pages. 9 illus. A Problem Seminar by Donald J. Newman 1982. viii, 113 pages. Problem-Solving Through Problems by Loren C. Larson 1983. xi, 344 pages. 104 illus. Demography Through Problems by N. Keyfitz and J.A. Beekman 1984. viii, 141 pages. 22 illus. Problem Book for First Year Calculus by George W. B/uman 1984. xvi. 384 pages. 384 illus. Exercises in Integration by Claude George 1984. x. 550 pages. 6 illus. Exercises in Number Theory by D.P. Parent 1984. x. 541 pages. Problems in Geometry by Marcel Berger et al. 1984. viii. 266 pages. 244 illus. Claude George Exercises in Integration With 6 Illustrations Springer-Verlag New York Berlin Heidelberg Tokyo Translator J.M. Cole Claude George University de Nancy I UER Sciences Mathematiques Boite Postale 239 54506 Vandoeuvre les Nancy Cedex France 17 St. Mary's Mount Leybum, North Yorkshire DL8 5JB U.K. Editor Paul R. Halmos Department of Mathematics Indiana University Bloomington, IN 47405 U.S.A. AMS Classifications: OOA07, 26-01, 28-01 Library of Congress Cataloging in Publication Data George, Claude. Exercises in integration. (Problem books in mathematics) Translation of: Exercices et problemes d`int6gration. Bibliography: p. Includes indexes. 1. Integrals, Generalized-Problems, exercises, etc. I. Title. II. Series. QA312.G39513 1984 515.4 84-14036 Title of the original French edition: Exercices et problemes d'integration, © BORDAS, Paris, 1980. © 1984 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York, 10010, U.S.A. Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America. 987654321 ISBN 0-387-96060-0 Springer-Verlag New York Berlin Heidelberg Tokyo ISBN 3-540-96060-0 Springer-Verlag Berlin Heidelberg New York Tokyo Introduction Having taught the theory of integration for several years at the University of Nancy I, then at the Ecole des Mines of the same city, I had followed the custom of the times of writing up detailed solutions of exercises and problems, which I used to distribute to the students every week. Some colleagues who had had occasion to use these solutions have persuaded me that this work would be interesting to many students, teachers and researchers. The majority of these exercises are at the master's level; to them I have added a number directed to those who would wish to tackle greater difficulties or complete their knowledge on various points of the theory (third year students, diploma of education students, researchers, etc.). This book, I hope, will render to students the services that this kind of book brings them in general, with the reservation that can always be made in this case: that certain of them will be tempted to look at the solution to the exercises which are put to.them without any personal effort. There is hardly any need to emphasize that such a use of this book would be no benefit. the other hand, the student who On after having worked seriously upon a problem, seeks some pointers from the solution, or compares it with his own, will be using this work in the optimal way. V INTRODUCTION