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The closed-form integration of arbitrary functions

  • A.D. Fitt (a1) and G.T.Q. Hoare (a1)


Consider the usual experience of a student who progresses far enough in school mathematics to begin to study the calculus. After motivation of the topic of “rates of change”, simple differentiation of polynomials is learned. More advanced functions are then considered, and eventually the student meets the product, quotient and chain rules. The result: with enough algebraic accuracy and persistence, the student can determine derived functions for virtually any sufficiently wellbehaved combination of standard functions.



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1. Ritt, J.F., Integration in finite terms, Columbia University Press, New York (1948).
2. Davenport, J.H., Siret, Y. and Tournier, E., Computer algebra. Academic Press (1988).
3. Liouville, J.Sur la determination des integrales dont la valeur est algebraique”, J. Ecole Polytech. 14 124193 (1833).
4. Rosenlicht, M.Integration in finite terms”. Amer. Math. Month. 79 963972 (1972).
5. Mead, D.G.Integration”, Amer. Math. Month. 68 152156 (1961).
6. Risch, R.H.Implicitly elementary integrals”, Proc. Amer. Math. Soc. 57 17 (1976).
7. Ostrowski, A.Sur la integrabilite elementaire de quelques classes d’expressions”. Comm. Math. Helv. 18 283308 (1946).
8. Hardy, G.H. The integration of functions of a single variable, Cambridge University Press (1905).
9. Risch, R.H.The problem of integration in finite terms”. Trans. Amer. Math. Soc. 139 167189 (1969).

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The closed-form integration of arbitrary functions

  • A.D. Fitt (a1) and G.T.Q. Hoare (a1)


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