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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 factor rule in integration ： ウィキペディア英語版 constant factor rule in integration The constant factor rule in integration is a dual of the constant factor rule in differentiation, and is a consequence of the linearity of integration. It states that a constant factor within an integrand can be separated from the integrand and instead multiplied by the integral. For example, where k is a constant: $\backslash int\; k\; \backslash frac\; dx\; =\; k\; \backslash int\; \backslash frac\; dx.\; \backslash quad$ == Proof == Start by noticing that, from the definition of integration as the inverse process of differentiation: :$y\; =\; \backslash int\; \backslash frac\; dx.$ Now multiply both sides by a constant ''k''. Since ''k'' is a constant it is not dependent on ''x'': :$ky\; =\; k\; \backslash int\; \backslash frac\; dx.\; \backslash quad\; \backslash mbox$ Take the constant factor rule in differentiation: :$\backslash frac\; =\; k\; \backslash frac.$ Integrate with respect to ''x'': :$ky\; =\; \backslash int\; k\; \backslash frac\; dx.\; \backslash quad\; \backslash mbox$ Now from (1) and (2) we have: :$ky\; =\; k\; \backslash int\; \backslash frac\; dx$ :$ky\; =\; \backslash int\; k\; \backslash frac\; dx.$ Therefore: :$\backslash int\; k\; \backslash frac\; dx\; =\; k\; \backslash int\; \backslash frac\; dx.\; \backslash quad\; \backslash mbox$ Now make a new differentiable function: :$u\; =\; \backslash frac.$ Substitute in (3): :$\backslash int\; ku\; dx\; =\; k\; \backslash int\; u\; dx.$ Now we can re-substitute ''y'' for something different from what it was originally: :$y\; =\; u.\; \backslash ,$ So: :$\backslash int\; ky\; dx\; =\; k\; \backslash int\; y\; dx.$ This is the constant factor rule in integration. A special case of this, with ''k''=-1, yields: :$\backslash int\; -y\; dx\; =\; -\backslash int\; y\; dx.$ 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 factor rule in 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.