Chapter 4 | Applications of Derivatives
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d dx (sin
x ) =cos x ,
so F ( x ) = sin x is an antiderivative of cos x . Therefore, every antiderivative of cos x is of the form sin x + C for some constant C and every function of the form sin x + C is an antiderivative of cos x . d. Since d dx ( e x ) = e x , then F ( x ) = e x is an antiderivative of e x . Therefore, every antiderivative of e x is of the form e x + C for some constant C and every function of the form e x + C is an antiderivative of e x .
Find all antiderivatives of f ( x ) = sin x .
4.49
Indefinite Integrals We now look at the formal notation used to represent antiderivatives and examine some of their properties. These properties allow us to find antiderivatives of more complicated functions. Given a function f , we use the notation f ′( x ) or df dx to denote the derivative of f . Here we introduce notation for antiderivatives. If F is an antiderivative of f , we say that F ( x )+ C is the most general antiderivative of f and write ∫ f ( x ) dx = F ( x )+ C . The symbol ∫ is called an integral sign , and ∫ f ( x ) dx is called the indefinite integral of f .
Definition Given a function f , the indefinite integral of f , denoted ∫ f ( x ) dx , is the most general antiderivative of f . If F is an antiderivative of f , then ∫ f ( x ) dx = F ( x )+ C . The expression f ( x ) is called the integrand and the variable x is the variable of integration .
Given the terminology introduced in this definition, the act of finding the antiderivatives of a function f is usually referred to as integrating f . For a function f and an antiderivative F , the functions F ( x )+ C , where C is any real number, is often referred to as the family of antiderivatives of f . For example, since x 2 is an antiderivative of 2 x and any antiderivative of 2 x is of the form x 2 + C , we write
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