Quadratintegrierbare

Quadratintegrierbare is a German term used in mathematics to denote the property of being integrable by quadrature. It describes a function, equation, or problem whose solution can be obtained by a finite sequence of algebraic operations and quadratures, i.e., evaluations of definite or indefinite integrals. In practice, solving by quadrature means reducing a problem to a limited number of steps that involve basic algebraic manipulation and integration.

In the context of differential equations, a first-order equation dy/dx = F(x, y) is quadratintegrierbar if its general solution can be constructed through a finite number of such steps. A classic example is a separable equation, which leads to ∫ dy/g(y) = ∫ f(x) dx + C. More broadly, a solution that can be expressed using elementary functions together with a finite number of integrals qualifies as Liouvillian, i.e., obtainable by quadrature.

The concept is related to Liouville’s theorem on elementary antiderivatives and to differential Galois theory, which provides criteria for when antiderivatives can be obtained by quadrature. It distinguishes between solvability by quadrature and the existence of a purely elementary closed form: some problems yield antiderivatives that are not elementary but are still obtainable via a finite sequence of quadratures or are expressible in terms of integrals or special functions defined by integrals.

In modern usage, “integrable by quadrature” is more common in English-language literature, while “quadratintegrierbar” appears in older or German-language sources. See also quadrature, Liouville theory, and Liouvillian functions.