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单词 Differentiation under the integral sign
释义

Differentiation under the integral sign

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积分符号内取微分

积分符号内取微分是一个在数学的微积分领域中很有用的运算。它是说，假定

F(x)=\int_{a(x)}^{b(x)}f(x,t)\,dt,在\,x_0\leq x\leq x_1\,时

f(x,t)\, 与 \frac{\partial}{\partial x}\,f(x,t)\, 对t\, 和 x\, 在(t,x)\, 平面连续, a(x)\leq t\leq b(x)\,, x_0\leq x\leq x_1\,, 且若对于x_0\leq x\leq x_1\,, a(x)\, 与 b(x)\, 及其导数连续，那幺

\begin{align} \frac{d}{dx}\,F(x) &= \left(\frac{\partial F}{\partial b}\right)\frac{db}{dx} - \left(\frac{\partial F}{\partial a}\right)\frac{da}{dx} + \int_{a(x)}^{b(x)}\frac{\partial}{\partial x}f(x,t)\; dt \\ &= f(x,b(x))\,b'(x) - f(x,a(x))\,a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}\, f(x,t)\; dt\, \end{align}
当   \,x_0\leq x\leq x_1\,\,.
英语百科

Differentiation under the integral sign 积分符号内取微分

Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows:

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).
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更新时间：2025/6/22 14:43:07