This page was motivated by many discussions on the cyberboard regarding
Liouville first proved (in 1835) that if f (x) and g(x) are rational functions (where g(x) is not a constant), then f (x)eg(x)dx is elementary if and only if there exists a rational function R(x) such that
You should be able to use this theorem to easily show that e-x2dx is NOT elementary.
This theorem can also be used to prove that integrals like
See KASPER T. (1980): "Integration in Finite Terms: the Liouville Theory", Mathematics Magazine 53 pp 195 - 201.
See also papers by Maxwell Rosenlicht in the Pacific Journal of Mathematics 54 (1968) pp 153 - 161 and 65 (1976) pp 485 - 492.
Alternatively, Ritt (in his textbook Integration in Finite Terms, New York: Columbia University Press 1948) has the wicked result that if
f (x)eg(x)dx can be integrated in a finite number of terms using elementary functions,
then the primitive must be
R(x)eg(x) for some rational function R(x).
So by taking the derivatives of both sides of
Additionally, Chebyshev first proved that xp(a + bxr)qdx can be integrated in a finite number of elementary terms if and only if at least one of , q or + q is an integer.
See MARCHISOTO and ZAKERI (1994): "An Invitation to Integration in Finite Terms", The College Mathematics Journal 25 No. 4 Sept. pp 295 - 308.
A good reference on non-elementary integrals is MEAD D. G. (1961): "Integration", American Mathematical Monthly Feb. pp 152 - 156.
Another paper to look at is FITT A. D. and HOARE G. T. Q. (1993): "The Closed-Form integration of Arbitrary Functions" Mathematical Gazette pp 227 - 236.
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Author: Mr. Fantastic