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.