Nå er det viktig å presisere hva en mener med en integrerbar funksjon.
Vi skiller ofte mellom funksjoner som er integrerbare, og de som kan skrives som en elementær funksjon.
http://en.wikipedia.org/wiki/Elementary_function
In mathematics, an elementary function is a function of one variable built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations (+ – × ÷). By allowing these functions (and constants) to be complex numbers, trigonometric functions and their inverses become included in the elementary functions
Eksempelvis kan mange integraler skrives om til summer, eller annet som gjør at de kan beregnes til vilkårlig presisjon. Dilog, polylog, erf er stikkord her.
Spørsmålet blir da, hvilke integrerbare funksjoner kan uttrykkes som elementær funksjon? Det er faktisk vanskelig å finne slike funksjoner, da det i praksis er svært få funksjoner som i det hele tatt er integrerbare.
Noen metoder finnes dog. Bla Risz algoritmen. Sakser likegreit fra wikipedia.
http://en.wikipedia.org/wiki/Integral#S ... algorithms
A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist. For instance, it is known that the antiderivatives of the functions exp(x2), xx and (sin x)/x cannot be expressed in the closed form involving only rational and exponential functions, logarithm, trigonometric and inverse trigonometric functions, and the operations of multiplication and composition; in other words, none of the three given functions is integrable in elementary functions, which are the functions which may be built from rational functions, roots of a polynomial, logarithm, and exponential functions. The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and, if it is, to compute it. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule. Consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may be still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in Mathematica and other computer algebra systems, does just that for functions and antiderivatives built from rational functions, radicals, logarithm, and exponential functions.
Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the special functions of physics (like the Legendre functions, the hypergeometric function, the Gamma function, the Incomplete Gamma function and so on - see Symbolic integration for more details). Extending the Risch's algorithm to include such functions is possible but challenging and has been an active research subject.
http://en.wikipedia.org/wiki/Risch_algorithm
Dog er nok denne algoritmen noe hverdagsfolk ikke benytter seg av. Som det står greit
The Risch algorithm is summarized (in more than 100 pages) in Algorithms for Computer Algebra
Selv tar jeg det litt på magefølelse, og den sier at en funksjon ikke er integrerbar til det motsatte er bevist.