On the values of integrals with the variable taken from $x=0$ to $x=\infty$

TL;DR

This paper discusses the evaluation of integrals with variable limits from zero to infinity, focusing on properties of the gamma function and applications to the geometry of clothoids, originally translated from Euler's 1781 work.

Contribution

It provides historical insights and general results on integrals with infinite limits and gamma function properties, based on Euler's original analysis.

Findings

Proved general properties of the gamma function.

Analyzed the endpoint of a clothoid spiral.

Connected integral values to geometric applications.

Abstract

This is a translation from the Latin original, "De valoribus integralium a termino variabilis x=0 usque ad x=infinity extensorum" (1781). This is E675 in the Enestrom index. Euler wants to find the location of the end point of a clothoid, a type of spiral. He proves some general results about the gamma function.