Review_Intergation1_M2420 - 1 Antiderivatives and Indenite...

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1 Antiderivatives and Inde°nite Integral De°nition 1.1 Let I denote an interval in R and f : I ! R be a function. We say that a function F : I ! R is an antiderivative of f ( x ) on I if F 0 ( x ) = f ( x ) for all x 2 I; or equivalently, using the notion of the di/erential dF ( x ) = f ( x ) dx for all x 2 I: Proposition 1.2 Let F 1 : I ! R and F 2 : I ! R be two antiderivatives of f : I ! R . Then, there exists a constant C 2 R such that F 1 ( x ) = F 2 ( x ) + C for all x 2 I . Proof. Since the function ( x ) = F 1 ( x ) ° F 2 ( x ) ; x 2 I; has the derivative 0 ( x ) = 0 , it follows that ( x ) is a constant function, i.e. there exists C 2 R such that ( x ) = F 1 ( x ) ° F 2 ( x ) = C and the conclusion follows. It is clear, by Proposition 1.2, that in order to °nd all antiderivatives of a given function f ( x ) on an interval I , it is su¢ cient to °nd just one antiderivative F ( x ) and any other antiderivative will be equal to F ( x ) + C for some constant C . The expression F ( x ) + C , where C denotes an arbitrary constant, is called the inde°nite integral of f ( x ) and is denoted by Z f ( x ) dx: It is convenient to think about the inde°nite integral Z f ( x ) dx = F ( x ) + C as a parameterized family (where C is a parameter) of all antiderivatives of the function f ( x ) . However, this explanation is correct only when the function f ( x ) is de°ned on an interval. For example the function F ( x ) = ° 1 2 x 2 + C is the inde°nite integral of f ( x ) = 1 x 3 ; but as we can see the function G ( x ) = ( ° 1 2 x 2 + 1 for x < 0; ° 1 2 x 2 ° 1 for x > 0 ; is also an antiderivative of f ( x ) which can not be expressed as F ( x )+ C with one value of the constant C . Therefore, we must ±improve²this explanation of the de°nite integral by adding that R f ( x ) dx denotes the family of all antiderivatives on intervals contained in the domain of f ( x ) . That means, if we know the formula of R f ( x ) dx = F ( x ) + C , then on every interval I contained in Dom ( f ) \ Dom ( F ) , every antiderivative of f ( x ) can be expressed in a form F ( x ) + C for a certain value of constant C . The symbol R can be easily explained in a context of de°nite integrals (assuming that we know Fundamental Theorem of Calculus). The symbol dx , which is associated with the notion of di/erential 1
indicates that x is the independent variable of the function f ( x ) and that the derivative of F ( x ) with respect to this variable should be equal to f ( x ) . On the other hand, R can be considered as an operator³ called integral or inde°nite integral which is the inverse operator of the operator of di/erentiation d dx , i.e. d dx Z f ( x ) dx = f ( x ) : However, if we think formally, the operator R should be considered rather as the inverse operator d associated with the operation of °nding a di/erential. Indeed, we have that d °Z f ( x ) dx ± = f ( x ) dx and Z df ( x ) dx = f ( x ) + C; so indeed, in a formal way the symbols d and R cancel each other out. The process of °nding the inde°nite integral of a certain function f ( x ) is called integration and the function f ( x ) is called the integrant . There is a fundamental question related to the above de°ned operation of integration. Which exactly are the functions f ( x ) for which an antiderivative F ( x ) exists, i.e. the inde°nite integral

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