To see the original (French) of this paper use the link and search for E521 at the Euler archive. The work was published in The Memoires de l'Academie des Sciences de Paris 1778, 1781, pp. 603-614. It also appears in the collected works of Euler, Opera Omnia: Series 1, Volume 18, pp. 69 - 82. The paper consists of extracts of letters written by Euler to Condorcet. The letters are about a couple of integrals and a series related to the binomial coefficients.
Euler evaluates the integrals by finding a function Q(x,y) such that integrating Q first with respect to x, then y, gives the integral he wants to evaluate. He then reverses the order of integration in order to evaluate the integral. Then he shows how he originally obtained the result through clever series manipulation. Finally he demonstrates a result about the sum of the squares of the binomial coefficients and uses this to obtain a series converging to four over pi by allowing fractions to interpolate his binomial result.