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Lists Of Integrals
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives. Historical development of integrals A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician (aka ) in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David Bierens de Haan for his ''Tables d'intégrales définies'', supplemented by ''Supplément aux tables d'intégrales définies'' in ca. 1864. A new edition was published in 1867 under the title ''Nouvelles tables d'intégrales définies''. These tables, which contain mainly integrals of elementary functions, remained in u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, log, or ''x''1/''n''). All elementary functions are continuous on their domains. Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s. Examples Basic examples Elementary functions of a single variable include: * Constant functions: 2,\ \pi,\ e, etc. * Rational powers of : x,\ x^2,\ \sqrt\ (x^\frac),\ x^\frac, etc. * more general algebraic functions: f(x) satisfying f(x)^5+f(x)+x=0, which is not expressible through n-th roots or rational powers of alone * Exponential functions: e^x, \ a^x * Logarithms: \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Izrail Solomonovich Gradshteyn
''Gradshteyn and Ryzhik'' (''GR'') is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is ''Table of Integrals, Series, and Products''. Since its first publication in 1943, it was considerably expanded and it soon became a "classic" and highly regarded reference for mathematicians, scientists and engineers. After the deaths of the original authors, the work was maintained and further expanded by other editors. At some stage a German and English dual-language translation became available, followed by Polish, English-only and Japanese versions. After several further editions, the Russian and German-English versions went out of print and have not been updated after the fall of the Iron Curtain, but the English version is still being actively maintained and refined by new editors, and it has recently been retranslated back into Russian as well. Overview One of the valuab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Gaussian Functions
In the expressions in this article, :\phi(x) = \frace^ is the standard normal probability density function, :\Phi(x) = \int_^x \phi(t) \, dt = \frac\left(1 + \operatorname\left(\frac\right)\right) is the corresponding cumulative distribution function (where erf is the error function) and : T(h,a) = \phi(h)\int_0^a \frac \, dx is Owen's T function. Owen has an extensive list of Gaussian-type integrals; only a subset is given below. Indefinite integrals :\int \phi(x) \, dx = \Phi(x) + C :\int x \phi(x) \, dx = -\phi(x) + C :\int x^2 \phi(x) \, dx = \Phi(x) - x\phi(x) + C :\int x^ \phi(x) \, dx = -\phi(x) \sum_^k \fracx^ + C lists this integral above without the minus sign, which is an error. See calculation bWolframAlpha/ref> :\int x^ \phi(x) \, dx = -\phi(x)\sum_^k\fracx^ + (2k+1)!!\,\Phi(x) + C In these integrals, ''n''!! is the double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Logarithmic Functions
The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. ''Note:'' ''x'' > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity. Integrals involving only logarithmic functions : \int\log_a x\,dx = x\log_a x - \frac = \frac : \int\ln(ax)\,dx = x\ln(ax) - x : \int\ln (ax + b)\,dx = \frac : \int (\ln x)^2\,dx = x(\ln x)^2 - 2x\ln x + 2x : \int (\ln x)^n\,dx = x\sum^_(-1)^ \frac(\ln x)^k : \int \frac = \ln, \ln x, + \ln x + \sum^\infty_\frac : \int \frac = \operatorname(x), the logarithmic integral. : \int \frac = -\frac + \frac\int\frac \qquad\mboxn\neq 1\mbox : \int \ln f(x)\,dx = x\ln f(x) - \int x\frac\,dx \qquad\mbox f(x) > 0\mbox Integrals involving logarithmic and power functions : \int x^m\ln x\,dx = x^\left(\frac-\frac\right) \qquad\mboxm\neq -1\mbox : \int x^m (\ln x)^n\,dx = \frac - \frac\int x^m (\ln x)^ dx \qquad\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Exponential Functions The following is a list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals. Indefinite integral Indefinite integrals are antiderivative functions. A constant (the constant of integrat |
