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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 Differentiation under the integral sign ： ウィキペディア英語版 Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows::Theorem. Let ''f''(''x'', ''t'') be a function such that both ''f''(''x'', ''t'') and its partial derivative ''fx''(''x'', ''t'') are continuous in ''t'' and ''x'' in some region of the (''x'', ''t'')-plane, including ''a''(''x'') ≤ ''t'' ≤ ''b''(''x''), ''x''0 ≤ ''x'' ≤ ''x''1. Also suppose that the functions ''a''(''x'') and ''b''(''x'') are both continuous and both have continuous derivatives for ''x''0 ≤ ''x'' ≤ ''x''1. Then for ''x''0 ≤ ''x'' ≤ ''x''1:::\fracx} \left (\int_^f(x,t)\,\mathrmt \right) = f(x,b(x))\cdot b'(x) - f(x,a(x))\cdot a'(x) + \int_^ \fracf(x,t)\; \mathrmt.This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just a particular case of the above formula, for ''a''(''x'') = ''a'', a constant, ''b''(''x'') = ''x'' and ''f''(''x'', ''t'') = ''f''(''t'').If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation::''ItDx'' = ''DxIt'',where ''Dx'' is the partial derivative with respect to ''x'' and ''It'' is the integral operator with respect to ''t'' over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.The following three basic theorems on the interchange of limits are essentially equivalent:* the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule)* the change of order of partial derivatives* the change of order of integration (integration under the integral sign; i.e., Fubini's theorem)== Higher dimensions == The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem::\fract} \int_ F(\vec, t) \,\mathrmV = \int_ \frac \,F(\vec, t)\,\mathrmV + \int_ \,F(\vec, t)\, \vec_b \cdot \mathrm\mathbfwhere F(\vec, t)\, is a scalar function, ''D''(''t'') and ∂''D''(''t'') denote a time-varying connected region of R3 and its boundary, respectively, \vec_b\, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and dΣ = n dS is the unit normal component of the surface element.The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in ''p''-dimensions is:,:\fract}\int_\omega=\int_ i__x\omega)+\int_ i_\dot,\,where Ω(''t'') is a time-varying domain of integration, ω is a ''p''-form, \vec\, is the vector field of the velocity, \vec=\frac\,, ''i'' denotes the interior product, ''dx''ω is the exterior derivative of ω with respect to the space variables only and \dot\, is the time-derivative of ω.== Proof of Theorem ==:Lemma. One has:::\frac \left (\int_a^b f(x)\; \mathrmx \right ) = f(b), \qquad \frac \left (\int_a^b f(x)\; \mathrmx \right )= -f(a).Proof. From proof of the fundamental theorem of calculus,:\begin \frac \left (\int_a^b f(x)\; \mathrmx \right ) &= \lim_ \frac \left(\int_a^ f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_b^ f(x)\,\mathrmx \\ &= \lim_ \frac \left(f(b) \Delta b + \mathcal\left(\Delta b^2\right) \right ) \\ &= f(b) \\ \frac \left (\int_a^b f(x)\; \mathrmx \right )&= \lim_ \frac \left(\int_^b f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_^a f(x)\,\mathrmx \\ &= \lim_ \frac \left(-f(a)\, \Delta a + \mathcal\left(\Delta a^2\right) \right )\\ &= -f(a).\endSuppose ''a'' and ''b'' are constant, and that ''f''(''x'') involves a parameter α which is constant in the integration but may vary to form different integrals. Assuming that ''f''(''x'', α) is a continuous function of ''x'' and α in the compact set , and that the partial derivative ''f''α(''x'', α) exists and is continuous, then if one defines::\varphi(\alpha) = \int_a^b f(x,\alpha)\;\mathrmx.:\varphi may be differentiated with respect to α by differentiating under the integral sign; i.e.,:\frac=\int_a^b\frac\,f(x,\alpha)\,\mathrmx.\,By the Heine–Cantor theorem it is uniformly continuous in that set. In other words for any ε > 0 there exists Δα such that for all values of ''x'' in (''b'' )::|f(x,\alpha+\Delta \alpha)-f(x,\alpha)|On the other hand::\begin\Delta\varphi &=\varphi(\alpha+\Delta \alpha)-\varphi(\alpha) \\&=\int_a^b f(x,\alpha+\Delta\alpha)\;\mathrmx - \int_a^b f(x,\alpha)\; \mathrmx \\&=\int_a^b \left (f(x,\alpha+\Delta\alpha)-f(x,\alpha) \right )\;\mathrmx \\&\leq \varepsilon (b-a)\endHence φ(α) is a continuous function.Similarly if \frac\,f(x,\alpha) exists and is continuous, then for all ε > 0 there exists Δα such that::\forall x \in (b ) \quad \left|\frac - \frac\right|Therefore,:\frac=\int_a^b\frac\;\mathrmx = \int_a^b \frac\,\mathrmx + Rwhere:|R| Now, ε → 0 as Δα → 0, therefore,:\lim_ \rarr 0}\frac= \frac = \int_a^b \frac\,f(x,\alpha)\,\mathrmx.\,This is the formula we set out to prove.Now, suppose:\int_a^b f(x,\alpha)\;\mathrmx=\varphi(\alpha),where ''a'' and ''b'' are functions of α which take increments Δ''a'' and Δ''b'', respectively, when α is increased by Δα. Then,:\begin\Delta\varphi &=\varphi(\alpha+\Delta\alpha)-\varphi(\alpha) \\&=\int_^f(x,\alpha+\Delta\alpha)\;\mathrmx\,-\int_a^b f(x,\alpha)\;\mathrmx\, \\&=\int_^af(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^bf(x,\alpha+\Delta\alpha)\;\mathrmx+\int_b^f(x,\alpha+\Delta\alpha)\;\mathrmx -\int_a^b f(x,\alpha)\;\mathrmx \\&=-\int_a^\,f(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^b()\;\mathrmx+\int_b^\,f(x,\alpha+\Delta\alpha)\;\mathrmx.\endA form of the mean value theorem, \int_a^bf(x)\;\mathrmx=(b-a)f(\xi), where ''a'' :\Delta\varphi=-\Delta a\,f(\xi_1,\alpha+\Delta\alpha)+\int_a^b()\;\mathrmx+\Delta b\,f(\xi_2,\alpha+\Delta\alpha).Dividing by Δα, letting Δα → 0, noticing ξ1 → ''a'' and ξ2 → ''b'' and using the above derivation for:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmxyields:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmx+f(b,\alpha)\frac-f(a,\alpha)\frac. This is the general form of the Leibniz integral rule. Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows: :Theorem. Let ''f''(''x'', ''t'') be a function such that both ''f''(''x'', ''t'') and its partial derivative ''fx''(''x'', ''t'') are continuous in ''t'' and ''x'' in some region of the (''x'', ''t'')-plane, including ''a''(''x'') ≤ ''t'' ≤ ''b''(''x''), ''x''0 ≤ ''x'' ≤ ''x''1. Also suppose that the functions ''a''(''x'') and ''b''(''x'') are both continuous and both have continuous derivatives for ''x''0 ≤ ''x'' ≤ ''x''1. Then for ''x''0 ≤ ''x'' ≤ ''x''1: ::$\backslash fracx\}\; \backslash left\; (\backslash int\_^f(x,t)\backslash ,\backslash mathrmt\; \backslash right)\; =\; f(x,b(x))\backslash cdot\; b\text{'}(x)\; -\; f(x,a(x))\backslash cdot\; a\text{'}(x)\; +\; \backslash int\_^\; \backslash fracf(x,t)\backslash ;\; \backslash mathrmt.$ This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just a particular case of the above formula, for ''a''(''x'') = ''a'', a constant, ''b''(''x'') = ''x'' and ''f''(''x'', ''t'') = ''f''(''t''). If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation: :''ItDx'' = ''DxIt'', where ''Dx'' is the partial derivative with respect to ''x'' and ''It'' is the integral operator with respect to ''t'' over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule. The following three basic theorems on the interchange of limits are essentially equivalent: * the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule) * the change of order of partial derivatives * the change of order of integration (integration under the integral sign; i.e., Fubini's theorem) == Higher dimensions == The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem: :$\backslash fract\}\; \backslash int\_\; F(\backslash vec,\; t)\; \backslash ,\backslash mathrmV\; =\; \backslash int\_\; \backslash frac\; \backslash ,F(\backslash vec,\; t)\backslash ,\backslash mathrmV\; +\; \backslash int\_\; \backslash ,F(\backslash vec,\; t)\backslash ,\; \backslash vec\_b\; \backslash cdot\; \backslash mathrm\backslash mathbf$ where $F(\backslash vec,\; t)\backslash ,$ is a scalar function, ''D''(''t'') and ∂''D''(''t'') denote a time-varying connected region of R3 and its boundary, respectively, $\backslash vec\_b\backslash ,$ is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and dΣ = n dS is the unit normal component of the surface element. The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in ''p''-dimensions is:〔,〕 :$\backslash fract\}\backslash int\_\backslash omega=\backslash int\_\; i\_\_x\backslash omega)+\backslash int\_\; i\_\backslash dot,\backslash ,$ where Ω(''t'') is a time-varying domain of integration, ω is a ''p''-form, $\backslash vec\backslash ,$ is the vector field of the velocity, $\backslash vec=\backslash frac\backslash ,$, ''i'' denotes the interior product, ''dx''ω is the exterior derivative of ω with respect to the space variables only and $\backslash dot\backslash ,$ is the time-derivative of ω. == Proof of Theorem == :Lemma. One has: ::$\backslash frac\; \backslash left\; (\backslash int\_a^b\; f(x)\backslash ;\; \backslash mathrmx\; \backslash right\; )\; =\; f(b),\; \backslash qquad\; \backslash frac\; \backslash left\; (\backslash int\_a^b\; f(x)\backslash ;\; \backslash mathrmx\; \backslash right\; )=\; -f(a).$ Proof. From proof of the fundamental theorem of calculus, :$\backslash begin$ \frac \left (\int_a^b f(x)\; \mathrmx \right ) &= \lim_ \frac \left(\int_a^ f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_b^ f(x)\,\mathrmx \\ &= \lim_ \frac \left(f(b) \Delta b + \mathcal\left(\Delta b^2\right) \right ) \\ &= f(b) \\ \frac \left (\int_a^b f(x)\; \mathrmx \right )&= \lim_ \frac \left(\int_^b f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_^a f(x)\,\mathrmx \\ &= \lim_ \frac \left(-f(a)\, \Delta a + \mathcal\left(\Delta a^2\right) \right )\\ &= -f(a). \end Suppose ''a'' and ''b'' are constant, and that ''f''(''x'') involves a parameter α which is constant in the integration but may vary to form different integrals. Assuming that ''f''(''x'', α) is a continuous function of ''x'' and α in the compact set , and that the partial derivative ''f''α(''x'', α) exists and is continuous, then if one defines: :$\backslash varphi(\backslash alpha)\; =\; \backslash int\_a^b\; f(x,\backslash alpha)\backslash ;\backslash mathrmx.$ :$\backslash varphi$ may be differentiated with respect to α by differentiating under the integral sign; i.e., :$\backslash frac=\backslash int\_a^b\backslash frac\backslash ,f(x,\backslash alpha)\backslash ,\backslash mathrmx.\backslash ,$ By the Heine–Cantor theorem it is uniformly continuous in that set. In other words for any ε > 0 there exists Δα such that for all values of ''x'' in (''b'' ): : On the other hand: :$\backslash begin$ \Delta\varphi &=\varphi(\alpha+\Delta \alpha)-\varphi(\alpha) \\ &=\int_a^b f(x,\alpha+\Delta\alpha)\;\mathrmx - \int_a^b f(x,\alpha)\; \mathrmx \\ &=\int_a^b \left (f(x,\alpha+\Delta\alpha)-f(x,\alpha) \right )\;\mathrmx \\ &\leq \varepsilon (b-a) \end Hence φ(α) is a continuous function. Similarly if $\backslash frac\backslash ,f(x,\backslash alpha)$ exists and is continuous, then for all ε > 0 there exists Δα such that: : Therefore, :$\backslash frac=\backslash int\_a^b\backslash frac\backslash ;\backslash mathrmx\; =\; \backslash int\_a^b\; \backslash frac\backslash ,\backslash mathrmx\; +\; R$ where : Now, ε → 0 as Δα → 0, therefore, :$\backslash lim\_\; \backslash rarr\; 0\}\backslash frac=\; \backslash frac\; =\; \backslash int\_a^b\; \backslash frac\backslash ,f(x,\backslash alpha)\backslash ,\backslash mathrmx.\backslash ,$ This is the formula we set out to prove. Now, suppose :$\backslash int\_a^b\; f(x,\backslash alpha)\backslash ;\backslash mathrmx=\backslash varphi(\backslash alpha),$ where ''a'' and ''b'' are functions of α which take increments Δ''a'' and Δ''b'', respectively, when α is increased by Δα. Then, :$\backslash begin$ \Delta\varphi &=\varphi(\alpha+\Delta\alpha)-\varphi(\alpha) \\ &=\int_^f(x,\alpha+\Delta\alpha)\;\mathrmx\,-\int_a^b f(x,\alpha)\;\mathrmx\, \\ &=\int_^af(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^bf(x,\alpha+\Delta\alpha)\;\mathrmx+\int_b^f(x,\alpha+\Delta\alpha)\;\mathrmx -\int_a^b f(x,\alpha)\;\mathrmx \\ &=-\int_a^\,f(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^b()\;\mathrmx+\int_b^\,f(x,\alpha+\Delta\alpha)\;\mathrmx. \end A form of the mean value theorem, $\backslash int\_a^bf(x)\backslash ;\backslash mathrmx=(b-a)f(\backslash xi),$ where ''a'' < ξ < ''b'', can be applied to the first and last integrals of the formula for Δφ above, resulting in :$\backslash Delta\backslash varphi=-\backslash Delta\; a\backslash ,f(\backslash xi\_1,\backslash alpha+\backslash Delta\backslash alpha)+\backslash int\_a^b()\backslash ;\backslash mathrmx+\backslash Delta\; b\backslash ,f(\backslash xi\_2,\backslash alpha+\backslash Delta\backslash alpha).$ Dividing by Δα, letting Δα → 0, noticing ξ1 → ''a'' and ξ2 → ''b'' and using the above derivation for :$\backslash frac\; =\; \backslash int\_a^b\backslash frac\backslash ,f(x,\backslash alpha)\backslash ,\backslash mathrmx$ yields :$\backslash frac\; =\; \backslash int\_a^b\backslash frac\backslash ,f(x,\backslash alpha)\backslash ,\backslash mathrmx+f(b,\backslash alpha)\backslash frac-f(a,\backslash alpha)\backslash frac.$ This is the general form of the Leibniz integral rule. 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「Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows::Theorem. Let ''f''(''x'', ''t'') be a function such that both ''f''(''x'', ''t'') and its partial derivative ''fx''(''x'', ''t'') are continuous in ''t'' and ''x'' in some region of the (''x'', ''t'')-plane, including ''a''(''x'') ≤ ''t'' ≤ ''b''(''x''), ''x''0 ≤ ''x'' ≤ ''x''1. Also suppose that the functions ''a''(''x'') and ''b''(''x'') are both continuous and both have continuous derivatives for ''x''0 ≤ ''x'' ≤ ''x''1. Then for ''x''0 ≤ ''x'' ≤ ''x''1:::\fracx} \left (\int_^f(x,t)\,\mathrmt \right) = f(x,b(x))\cdot b'(x) - f(x,a(x))\cdot a'(x) + \int_^ \fracf(x,t)\; \mathrmt.This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just a particular case of the above formula, for ''a''(''x'') = ''a'', a constant, ''b''(''x'') = ''x'' and ''f''(''x'', ''t'') = ''f''(''t'').If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation::''ItDx'' = ''DxIt'',where ''Dx'' is the partial derivative with respect to ''x'' and ''It'' is the integral operator with respect to ''t'' over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.The following three basic theorems on the interchange of limits are essentially equivalent:* the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule)* the change of order of partial derivatives* the change of order of integration (integration under the integral sign; i.e., Fubini's theorem)== Higher dimensions == The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem::\fract} \int_ F(\vec, t) \,\mathrmV = \int_ \frac \,F(\vec, t)\,\mathrmV + \int_ \,F(\vec, t)\, \vec_b \cdot \mathrm\mathbfwhere F(\vec, t)\, is a scalar function, ''D''(''t'') and ∂''D''(''t'') denote a time-varying connected region of R3 and its boundary, respectively, \vec_b\, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and dΣ = n dS is the unit normal component of the surface element.The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in ''p''-dimensions is:,:\fract}\int_\omega=\int_ i__x\omega)+\int_ i_\dot,\,where Ω(''t'') is a time-varying domain of integration, ω is a ''p''-form, \vec\, is the vector field of the velocity, \vec=\frac\,, ''i'' denotes the interior product, ''dx''ω is the exterior derivative of ω with respect to the space variables only and \dot\, is the time-derivative of ω.== Proof of Theorem ==:Lemma. One has:::\frac \left (\int_a^b f(x)\; \mathrmx \right ) = f(b), \qquad \frac \left (\int_a^b f(x)\; \mathrmx \right )= -f(a).Proof. From proof of the fundamental theorem of calculus,:\begin \frac \left (\int_a^b f(x)\; \mathrmx \right ) &= \lim_ \frac \left(\int_a^ f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_b^ f(x)\,\mathrmx \\ &= \lim_ \frac \left(f(b) \Delta b + \mathcal\left(\Delta b^2\right) \right ) \\ &= f(b) \\ \frac \left (\int_a^b f(x)\; \mathrmx \right )&= \lim_ \frac \left(\int_^b f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_^a f(x)\,\mathrmx \\ &= \lim_ \frac \left(-f(a)\, \Delta a + \mathcal\left(\Delta a^2\right) \right )\\ &= -f(a).\endSuppose ''a'' and ''b'' are constant, and that ''f''(''x'') involves a parameter α which is constant in the integration but may vary to form different integrals. Assuming that ''f''(''x'', α) is a continuous function of ''x'' and α in the compact set , and that the partial derivative ''f''α(''x'', α) exists and is continuous, then if one defines::\varphi(\alpha) = \int_a^b f(x,\alpha)\;\mathrmx.:\varphi may be differentiated with respect to α by differentiating under the integral sign; i.e.,:\frac=\int_a^b\frac\,f(x,\alpha)\,\mathrmx.\,By the Heine–Cantor theorem it is uniformly continuous in that set. In other words for any ε > 0 there exists Δα such that for all values of ''x'' in (''b'' )::|f(x,\alpha+\Delta \alpha)-f(x,\alpha)|On the other hand::\begin\Delta\varphi &=\varphi(\alpha+\Delta \alpha)-\varphi(\alpha) \\&=\int_a^b f(x,\alpha+\Delta\alpha)\;\mathrmx - \int_a^b f(x,\alpha)\; \mathrmx \\&=\int_a^b \left (f(x,\alpha+\Delta\alpha)-f(x,\alpha) \right )\;\mathrmx \\&\leq \varepsilon (b-a)\endHence φ(α) is a continuous function.Similarly if \frac\,f(x,\alpha) exists and is continuous, then for all ε > 0 there exists Δα such that::\forall x \in (b ) \quad \left|\frac - \frac\right|Therefore,:\frac=\int_a^b\frac\;\mathrmx = \int_a^b \frac\,\mathrmx + Rwhere:|R| Now, ε → 0 as Δα → 0, therefore,:\lim_ \rarr 0}\frac= \frac = \int_a^b \frac\,f(x,\alpha)\,\mathrmx.\,This is the formula we set out to prove.Now, suppose:\int_a^b f(x,\alpha)\;\mathrmx=\varphi(\alpha),where ''a'' and ''b'' are functions of α which take increments Δ''a'' and Δ''b'', respectively, when α is increased by Δα. Then,:\begin\Delta\varphi &=\varphi(\alpha+\Delta\alpha)-\varphi(\alpha) \\&=\int_^f(x,\alpha+\Delta\alpha)\;\mathrmx\,-\int_a^b f(x,\alpha)\;\mathrmx\, \\&=\int_^af(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^bf(x,\alpha+\Delta\alpha)\;\mathrmx+\int_b^f(x,\alpha+\Delta\alpha)\;\mathrmx -\int_a^b f(x,\alpha)\;\mathrmx \\&=-\int_a^\,f(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^b()\;\mathrmx+\int_b^\,f(x,\alpha+\Delta\alpha)\;\mathrmx.\endA form of the mean value theorem, \int_a^bf(x)\;\mathrmx=(b-a)f(\xi), where ''a'' :\Delta\varphi=-\Delta a\,f(\xi_1,\alpha+\Delta\alpha)+\int_a^b()\;\mathrmx+\Delta b\,f(\xi_2,\alpha+\Delta\alpha).Dividing by Δα, letting Δα → 0, noticing ξ1 → ''a'' and ξ2 → ''b'' and using the above derivation for:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmxyields:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmx+f(b,\alpha)\frac-f(a,\alpha)\frac. This is the general form of the Leibniz integral rule.」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.