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

  • Dennis M. CatesEmail author
Chapter

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
$$\begin{aligned} \int _{x_0}^X{f(x) dx}=\int _{x_0}^{x_1}{f(x) dx}+\int _{x_1}^{x_2}{f(x) dx}+\cdots +\int _{x_{n-1}}^X{f(x) dx}. \end{aligned}$$

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sun CityUSA

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