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Nonelementary integral. In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. [1] A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. [2] This theorem also provides a basis for the ...
More detail may be found on the following pages for the lists of integrals: Gradshteyn, Ryzhik, Geronimus, Tseytlin, Jeffrey, Zwillinger, and Moll 's (GR) Table of Integrals, Series, and Products contains a large collection of results. An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with ...
Trigonometric integral. Plot of the hyperbolic sine integral function Shi (z) in the complex plane from −2 − 2i to 2 + 2i. Si (x) (blue) and Ci (x) (green) shown on the same plot. Integral sine in the complex plane, plotted with a variant of domain coloring. Integral cosine in the complex plane.
t. e. In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting . This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the ...
Liouville's theorem (differential algebra) In mathematics, Liouville's theorem, originally formulated by French mathematician Joseph Liouville in 1833 to 1841, [1][2][3] places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be ...
Fubini's theorem. In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is Lebesgue integrable on a rectangle , then one can evaluate the double integral as an ...
Calculus. In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards."
Simpson's 1/3 rule. Simpson's 1/3 rule, also simply called Simpson's rule, is a method for numerical integration proposed by Thomas Simpson. It is based upon a quadratic interpolation and is the composite Simpson's 1/3 rule evaluated for . Simpson's 1/3 rule is as follows: where is the step size for .
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