OF DEFINITE INTEGRALS WHOSE VALUES ARE INFINITE OR INDETERMINATE. PRINCIPAL VALUES OF INDETERMINATE INTEGRALS

Dennis M. Cates ()

Chapter Chapter 24 in Cauchy's Calcul Infinitésimal, 2019, pp 129-132 from Springer

Abstract: Abstract In the previous lectures, we have demonstrated several remarkable properties of the definite integral $$\begin{aligned} \int _{x_0}^X{f(x) dx}, \end{aligned}$$ but by supposing: $$ 1^{\circ } $$ that the limits $$ x_0, X $$ were finite quantities; $$ 2^{\circ } $$ that the function f(x) would remain finite and continuous between these same limits. When these two types of conditions are found fulfilled, then, in designating by $$ x_1, x_2, $$ $$ \dots , x_{n-1} $$ new values of x interposed between the extreme values $$ x_0, X, $$ we have $$\begin{aligned} \int _{x_0}^X{f(x) dx}=\int _{x_0}^{x_1}{f(x) dx}+\int _{x_1}^{x_2}{f(x) dx}+\cdots +\int _{x_{n-1}}^X{f(x) dx}. \end{aligned}$$

Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-11036-9_24

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DOI: 10.1007/978-3-030-11036-9_24

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