CALCULUS Integralis

CALCULUS Integralis, or Summatorius, is a method of integrating, or summing up moments, or differential

quantities; i. e. from a differential quantity given, to find the quantity from whose differencing the given differential results. Calculus.

The integral calculus, therefore, is the inverse of the differential one: whence the English, who usually call the differential method fluxions, give this calculus, which ascends from the fluxions, to the flowing or variable quantities: or as foreigners express it, from the differences to the sums, by the name of the inverse method of fluxions.

Hence, the integration is known to be justly performed, if the quantity found, according to the rules of the differential calculus, being differenced, produce that proposed to be summed.

Suppose $\backslash int$ the sign of the sum, or integral quantity, then $\backslash int\; y\; dx$ will denote the sum, or integral of the differential $y\; dx$.

To integrate, or sum up a differential quantity: it is demonstrated, first, that $\backslash int\; dx\; =\; x$; secondly, $\backslash int\; (dx\; +\; dy)\; =\; x\; +\; y$; thirdly, $\backslash int\; (x\; dy\; +\; y\; dx)\; =\; xy$; fourthly, $\backslash int\; (m\; x^n\; -\; x^m\; dx)\; =\; x\; m$; fifthly, $\backslash int\; (n\; :\; m)\; x^\{\backslash frac\{n-m\}\{m\}\}\; dx\; =\; x^\{\backslash frac\{n\}\{m\}\}$; sixthly, $\backslash int\; (y\; dx\; -\; x\; dy)\; :\; y^2\; =\; x\; :\; y$. Of these, the fourth and fifth cases are the most frequent, wherein the differential quantity is integrated, by adding a variable unity to the exponent, and dividing the sum by the new exponent multiplied into the differential of the root; v. g. the fourth case, by $m\; =\; (1+1)\; dx$, i. e. by $2\; dx$.

If the differential quantity to be integrated doth not come under any of these formulas, it must either be reduced to an integral finite, or an infinite series, each of whose terms may be summed.

It may be here observed, that, as in the analysis of finites, any quantity may be raised to any degree of power; but vice versa, the root cannot be extracted out of any number required; so in the analysis of infinites, any variable or flowing quantity may be differenced; but vice versa, any differential cannot be integrated. And as, in the analysis of finites, we are not yet arrived at a method of extracting the roots of all equations, so neither has the integral calculus arrived at its perfection: and as in the former we are obliged to have recourse to approximation, so in the latter we have recourse to infinite series, where we cannot attain to a perfect integration.