The closure under integration of the set of the elementary functions is the set of the Liouvillian functions. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.Įxamples of functions with nonelementary antiderivatives include: A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field operations). In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. JSTOR ( December 2009) ( Learn how and when to remove this template message).Unsourced material may be challenged and removed.įind sources: "Nonelementary integral" – news Use the Taylor series for e-x2 to evaluate the integral I integral03 4e-x2 dx I sigmak0infinity (-1)k/2k + 1 4 middit 32k + 1 I sigmak0infinity 1/k 4 middit 32k I sigmak0infinity 1/k (2k + 1) 4 middot 32k+1 I sigmak0infinity (-1)k/k (2k + 1) 4 middot 32k+1 I sigmak0. Please help improve this article by adding citations to reliable sources. This article needs additional citations for verification.
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