Speculationes super formula integrali {\int} (x^ndx)/{\surd}(aa-2bx+cxx), ubi simul egregiae observationes circa fractiones continuas occurrunt

TL;DR

Euler's work explores integrals involving quadratic expressions, establishing recursive relationships and deriving continued fractions for logarithmic and arctangent functions, with a translation from Latin to German.

Contribution

The paper introduces recursive formulas for specific integrals and derives continued fractions for key functions, expanding on Euler's original analysis.

Findings

Recursive relationships between integrals are established.

Continued fractions for log and arctan functions are derived.

The translation makes Euler's original work accessible to a modern audience.

Abstract

Euler evaluates the integrals in the title and recognizes a recursion between them, which he then uses to give continued fractions for the log and arctan. The paper is translated from Euler's Latin original into German.