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# ON THE TRANSITION OF INDEFINITE INTEGRALS TO DEFINITE INTEGRALS.

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}

## Copyright information

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

1. 1.Sun CityUSA