14 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L Chen X T Duong and Macquarie University 1999 This work is available free in the hope

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ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c ° W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 14 INTRODUCTION TO INTEGRATION 14.1. Antiderivatives In this chapter, we discuss the inverse process of diFerentiation. In other words, given a function f ( x ), we wish to ±nd a function F ( x ) such that F 0 ( x )= f ( x ). Any such function F ( x ) is called an antiderivative, or inde±nite integral, of the function f ( x ), and we write F ( x Z f ( x )d x. A ±rst observation is that the antiderivative, if it exists, is not unique. Suppose that the function F ( x ) is an antiderivative of the function f ( x ), so that F 0 ( x f ( x ). Let G ( x F ( x )+ C , where C is any ±xed real number. Then it is easy to see that G 0 ( x F 0 ( x f ( x ), so that G ( x ) is also an antiderivative of f ( x ). A second observation, somewhat less obvious, is that for any given function f ( x ), any two distinct antiderivatives of f ( x ) must diFer only by a constant. In other words, if F ( x ) and G ( x ) are both antiderivatives of f ( x ), then F ( x ) G ( x ) is a constant. In this chapter, we shall denote any such constant by C , with or without subscripts. An immediate consequence of this second observation is the following simple result related to the derivatives of constants in Section 11.1. ANTIDERIVATIVES OF ZERO. We have Z 0d x = C. In other words, the antiderivatives of the zero function are precisely all the constant functions. Indeed, many antiderivatives can be obtained simply by referring to various rules concerning deriva- tives. We list here a number of such results. The ±rst of these is related to the constant multiple rule for diFerentiation in Section 11.2. This chapter was written at Macquarie University in 1999.
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14–2 W W L Chen and X T Duong : Elementary Mathematics CONSTANT MULTIPLE RULE. Suppose that a function f ( x ) has antiderivatives. Then for any Fxed real number c ,wehave Z cf ( x )d x = c Z f ( x x. ANTIDERIVATIVES OF POWERS. (a) Suppose that n is a Fxed real number such that n 6 = 1 . Then Z x n d x = 1 n +1 x n +1 + C. (b) We have Z x 1 d x = log | x | + C. Proof. Part (a) is a consequence of the rule concerning derivatives of powers in Section 11.1. If x> 0, then part (b) is a consequence of the rule concerning the derivative of the logarithmic function in Section 12.3. If x< 0, we can write | x | = u , where u = x . It then follows from the Chain rule that d d x (log | x | )= d u d x × d d u (log u 1 u = 1 x (1) again. Corresponding to the sum rule for diFerentiation in Section 11.2, we have the following. SUM RULE. Suppose that functions f ( x ) and g ( x ) have antiderivatives. Then Z ( f ( x )+ g ( x ))d x = Z f ( x x + Z g ( x x.
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14 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L Chen X T Duong and Macquarie University 1999 This work is available free in the hope

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