“differentiation under the integral sign”的概念、定义、翻译、参考文献-科学参考

differentiation under the integral sign

Mathematics

单词	differentiation under the integral sign
释义	differentiation under the integral sign Mathematics This states that $\frac{d}{d α} \int_{a}^{b} f (t, α) d t = \int_{a}^{b} \frac{\partial f}{\partial α} (t, α) d t .$ This is a special case of Leibniz’s integral rule. As an example, for α‎ > 0 define $I (α) = \int_{0}^{\infty} \frac{e^{- x} - e^{- α x}}{x} d x .$ Then $I' (α) = \int_{0}^{\infty} \frac{x e^{- α x}}{x} d x = \int_{0}^{\infty} e^{- α x} d x = \frac{1}{α},$ and integrating gives I(α‎)= c + lnα‎ for some constant c. As I(1) = 0 then I(α‎) = lnα‎.
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