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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Constant of integration ： ウィキペディア英語版 Constant of integration In calculus, the indefinite integral of a given function (i.e., the set of all antiderivatives of the function) is only defined up to an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function $f(x)$ is defined on an interval and $F(x)$ is an antiderivative of $f(x)$, then the set of ''all'' antiderivatives of $f(x)$ is given by the functions $F(x)\; +\; C$, where ''C'' is an arbitrary constant. The constant of integration is sometimes omitted in lists of integrals for simplicity. ==Origin of the constant== The derivative of any constant function is zero. Once one has found one antiderivative $F(x)$ for a function $f(x)$, adding or subtracting any constant ''C'' will give us another antiderivative, because $(F(x)\; +\; C)\text{'}\; =\; F\backslash ,\text{'}(x)\; +\; C\backslash ,\text{'}\; =\; F\backslash ,\text{'}(x)$. The constant is a way of expressing that every function with at least one antiderivative has an infinite number of them. For example, suppose one wants to find antiderivatives of $\backslash cos(x)$. One such antiderivative is $\backslash sin(x)$. Another one is $\backslash sin(x)+1$. A third is $\backslash sin(x)-\backslash pi$. Each of these has derivative $\backslash cos(x)$, so they are all antiderivatives of $\backslash cos(x)$. It turns out that adding and subtracting constants is the only flexibility we have in finding different antiderivatives of the same function. That is, all antiderivatives are the same up to a constant. To express this fact for cos(''x''), we write: :$\backslash int\; \backslash cos(x)\backslash ,dx\; =\; \backslash sin(x)\; +\; C.$ Replacing ''C'' by a number will produce an antiderivative. By writing ''C'' instead of a number, however, a compact description of all the possible antiderivatives of cos(''x'') is obtained. ''C'' is called the constant of integration. It is easily determined that all of these functions are indeed antiderivatives of $\backslash cos(x)$: :$\backslash begin$ \frac(+ C ) &= \frac() + \frac() \\ &= \cos(x) + 0 \\ &= \cos(x) \end 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「Constant of integration」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.