An easy method for finding the integral of the formula $\int (x^{n+p} - 2 x^n\cos\zeta + x^{n-p})/(x^{2n} - 2 x^n\cos\theta + 1) dx/x$ when the upper limit of integration is $x=1$ or $x=\infty$

TL;DR

This paper revisits Euler's 1776 integral formula, providing a modern translation, historical context, and a list of interesting definite integrals and series derived from the original work, including Fourier and Laplace transforms.

Contribution

It offers a modern translation and analysis of Euler's original integral formula, highlighting its connections to Fourier and Laplace transforms and providing a comprehensive list of related integrals.

Findings

Euler's integral formula relates to Fourier and Laplace transforms.

The paper includes a list of interesting definite integrals and series derived from Euler's work.

Historical context and modern notation enhance understanding of Euler's original results.

Abstract

This is a translation of an article presented by Leonhard Euler on 18 March 1776 (Opera Omnia I-XVIII, pp. 265-290) and of summaries for it by Sim\'eon Denis Poisson in 1820 and by Heinrich Burkhardt in 1916. An appendix lists in modern notation interesting definite integrals and series which Euler, after using partial fractions to prove his main formula, obtained formally by allowing the parameters $n, p, zeta and theta$ to take particular, even pure imaginary, values : the Fourier cosine and two-sided Laplace transforms of the trigonometric and of the hyperbolic secant (also squared), and their partial fraction decomposition. The source archive provides, in Plain TeX, LaTeX and PDF formats, the corrected Latin text and a complete translation into French.