|
In calculus, an antiderivative, primitive function, primitive integral or indefinite integral〔 Antiderivatives are also called general integrals, and sometimes integrals. The latter term is generic, and refers not only to indefinite integrals (antiderivatives), but also to definite integrals. When the word integral is used without additional specification, the reader is supposed to deduce from the context whether it is referred to a definite or indefinite integral. Some authors define the indefinite integral of a function as the set of its infinitely many possible antiderivatives. Others define it as an arbitrarily selected element of that set. Wikipedia adopts the latter approach. 〕 of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically as ′ . The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. The discrete equivalent of the notion of antiderivative is antidifference. ==Example== The function ''F''(''x'') = ''x''3/3 is an antiderivative of ''f''(''x'') = ''x''2. As the derivative of a constant is zero, ''x''2 will have an infinite number of antiderivatives, such as ''x''3/3, ''x''3/3 + 1, ''x''3/3 - 2, etc. Thus, all the antiderivatives of ''x''2 can be obtained by changing the value of C in ''F''(''x'') = ''x''3/3 + ''C''; where ''C'' is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other; each graph's vertical location depending upon the value of ''C''. In physics, the integration of acceleration yields velocity plus a constant. The constant is the initial velocity term that would be lost upon taking the derivative of velocity because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion (position, velocity, acceleration, and so on). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「antiderivative」の詳細全文を読む スポンサード リンク
|