ON THE TRANSITION OF INDEFINITE INTEGRALS TO DEFINITE INTEGRALS

Dennis M. Cates ()

Chapter Chapter 32 in Cauchy's Calcul Infinitésimal, 2019, pp 173-178 from Springer

Abstract: 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 $$\begin{aligned} \varphi (x)+\int {\chi (x) \, dx}. \end{aligned}$$

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_32

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