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ON THE TRANSITION OF INDEFINITE INTEGRALS TO DEFINITE INTEGRALS.

  • Dennis M. CatesEmail author
Chapter

Abstract

To integrate the equation
$$\begin{aligned} dy=f(x) \, dx, \end{aligned}$$
or the differential expression \(f(x) \, dx\), starting from \(x=x_0\), is to find a continuous function of x which has the double property of giving for a differential, \(f(x) \, dx,\) and vanishing for \(x=x_0\). This function, before being included in the general formula
$$\begin{aligned} \int {f(x) \, dx}=\int _{x_0}^{x}{f(x) \, dx}+\mathscr {C}, \end{aligned}$$
will necessarily be reduced to the integral \(\int _{x_0}^{x}{f(x) \, dx}\), if the function f(x) is itself continuous with respect to x between the two limits of this integral. Conceive now that, the two functions \(\varphi (x)\) and \(\chi (x)\) being continuous between these limits, the general value of y derived from equation (1) is presented under the form
$$\begin{aligned} \varphi (x)+\int {\chi (x) \, dx}. \end{aligned}$$

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sun CityUSA

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