SINGULAR DEFINITE INTEGRALS.

Abstract

Consider that an integral relative to x, and in which the function under the \(\int \) sign is denoted by f(x), is taken between two limits infinitely close to a definite particular value a attributed to the variable \(x. \ \) If this value a is a finite quantity, and if the function f(x) remains finite and continuous in the neighborhood of \(x=a, \) then, by virtue of formula ( 19) (twenty-second lecture), the proposed integral will be essentially null. But, it can obtain a finite value different from zero or even an infinite value, if we have $$\begin{aligned} a=\displaystyle \frac{\pm }{\infty } \ \ \ \ \ \ \ \ \text {or else} \ \ \ \ \ \ \ \ f(a)=\pm \infty . \end{aligned}$$ In this latter case, the integral in question will become what we will call a singular definite integral. It will ordinarily be easy to calculate its value with the help of formulas ( 15) and ( 16) of the twenty-third lecture, as we shall see.