E695 -- Integratio succincta formulae integralis maxime memorabilis ∫ dz/((3±zz)*3√(1±3zz))

(A succinct integration of the most memorable integral formula ∫ dz/((3±zz)*3√(1±3zz)))


The integrals in the title are calculated with some very slick 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.

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