ABSTRACT
The Literature devoted to the approximate computation of definite in tegrals is extremely voluminous, so that these questions are the most de veloped ones in the science of approximate calculations. However, almost all the studies there are concerned exclusively with estimation of the error in the application of different classical formulas of numerical integration to integrals of functions with a definite number of derivatives in the interval of integration. At the same time the question of approximate calculation of improper integrals or proper integrals of functions with a first deriva tive that becomes infinite in the interval of integration or near it is hardly addressed. Nevertheless integrals of this kind often appear and the use of ordinary formulas for their computation is impossible since, if the integral is improper, one of the ordinates may become infinite, while if the integrand itself is bounded but possesses an infinite first derivative, then the error in the application of the formula becomes extremely large. The same effect appears if a singular point of the integrand is located outside the interval of integration but close to it.
