The discussion focuses on integrating the function ##\frac{1}{\sqrt{x^3 + x^2 + x + 1}} \, dx##, highlighting the use of elliptic integrals. Participants suggest rewriting the polynomial as ##(x^2+1)(x+1)## and using substitutions such as ##x^2+1=u##. The conversation emphasizes the complexity of the integral, noting that it falls outside standard calculus teachings and may require advanced techniques involving Legendre canonical forms. Resources like Wolfram Alpha are recommended for testing integral complexity.
PREREQUISITESMathematicians, advanced calculus students, and anyone interested in the complexities of integrating non-standard functions, particularly those involving elliptic integrals.
It's a pretty interesting subject, elliptic integrals and functions if you're into that sort of thing. Check Wikipedia article: Elliptic integralsaskor said:How do you integrate ##\frac{1}{\sqrt{x^3 + x^2 + x + 1}} \, dx##?
Please give me some hints and clues.
. . . , with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).