A succinct integration of the most memorable integral formula â« âz/((3Â±zz)∙(1Â±3zz)1/3)
Euler used a cube root in the title of this work, rather than the 1/3 power indicated in the Euler Archive's listed title.
The integrals in the title are calculated with some very clever substitutions and imaginative use of imaginaries. Seeing that this method is not very obvious, Euler proceeds to rederive the results (with fewer details) in a more natural way, but nonetheless judges the former method to be better. The paper concludes with some remarks affirming that no matter how one calculates them, one must use imaginary quantities.
Original Source Citation
Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 10, pp. 20-26.
Opera Omnia Citation
Series 1, Volume 19, pp.287-296.