Cauchy's Calcul Infinitésimal pp 173-178 | Cite as

# ON THE TRANSITION OF INDEFINITE INTEGRALS TO DEFINITE INTEGRALS.

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## Abstract

To integrate the equation or the differential expression \(f(x) \, dx\), will necessarily be reduced to the integral \(\int _{x_0}^{x}{f(x) \, dx}\), if the function

$$\begin{aligned} dy=f(x) \, dx, \end{aligned}$$

*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$$\begin{aligned} \int {f(x) \, dx}=\int _{x_0}^{x}{f(x) \, dx}+\mathscr {C}, \end{aligned}$$

*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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