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*)*e*^{g(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*^{-x2}*dx* 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*)*e*^{g(x)}*dx* can be integrated in a finite number of terms using elementary functions,
then the primitive must be
*R*(*x*)*e*^{g(x)} for some rational function *R*(*x*).

So by taking the derivatives of both sides of

Additionally, Chebyshev first proved that
*x*^{p}(*a* + *bx*^{r})^{q}*dx* 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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