Integration of Nonelementary Functions

This page was motivated by many discussions on the cyberboard regarding

$\displaystyle \int$e-x2dx

Liouville first proved (in 1835) that if f (x) and g(x) are rational functions (where g(x) is not a constant), then $ \int$f (x)eg(x)dx is elementary if and only if there exists a rational function R(x) such that

f (x) = R'(x) + R(x)g(x).

You should be able to use this theorem to easily show that $ \int$e-x2dx is NOT elementary.

This theorem can also be used to prove that integrals like

$\displaystyle \int$xxdx,    $\displaystyle \int$$\displaystyle {\frac{{\sqrt{\sin x}}}{{x}}}$dx,    $\displaystyle \int$$\displaystyle {\frac{{e^x}}{{x}}}$dx,    $\displaystyle \int$$\displaystyle {\frac{{\sin x}}{{x}}}$dx

are NOT elementary.

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 $ \int$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

$\displaystyle \int$f (x)e-x2dx = R(x)e-x2

and matching powers, you can show that $ \int$e-x2dx cannot be integrated in a finite number of elementary terms.

Additionally, Chebyshev first proved that $ \int$xp(a + bxr)qdx can be integrated in a finite number of elementary terms if and only if at least one of $\displaystyle {\frac{{p + 1}}{{r}}}$, q or $\displaystyle {\frac{{p + 1}}{{r}}}$ + 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

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