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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\mb ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically as . The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called ''differentiation'', which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as and . Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity and acce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Function
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of is , or . The logarithm of to ''base'' is denoted as , or without parentheses, , or even without the explicit base, , when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number as its base; its use is widespread in mathematics and physics, because of its very simple derivative. The binary logarithm uses base and is frequently used in computer science. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Of Integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives. More specifically, if a function f(x) is defined on an interval, and F(x) is an antiderivative of f(x), then the set of ''all'' antiderivatives of f(x) is given by the functions F(x) + C, where C is an arbitrary constant (meaning that ''any'' value of C would make F(x) + C a valid antiderivative). For that reason, the indefinite integral is often written as \int f(x) \, dx = F(x) + C, although the constant of integration might be sometimes omitted in lists of integrals for simplicity. Origin The derivative of any constant function is zero. Once one has found one antiderivative F(x) for a function f( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Integral
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value x. Integral representation The logarithmic integral has an integral representation defined for all positive real numbers ≠ 1 by the definite integral : \operatorname(x) = \int_0^x \frac. Here, denotes the natural logarithm. The function has a singularity at , and the integral for is interpreted as a Cauchy principal value, : \operatorname(x) = \lim_ \left( \int_0^ \frac + \int_^x \frac \right). Offset logarithmic integral The offset logarithmic integral or Eulerian logarithmic integral is defined as : \operatorname(x) = \int_2^x \frac = \operatorname(x) - \operatorname(2). As s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milton Abramowitz
Milton Abramowitz (19 February 1915 in Brooklyn, New York – 5 July 1958) was a Jewish American mathematician at the National Bureau of Standards who, with Irene Stegun, edited a classic book of mathematical tables called '' Handbook of Mathematical Functions'', widely known as ''Abramowitz and Stegun''. Abramowitz died of a heart attack in 1958, at which time the book was not yet completed but was well underway. Stegun took over management of the project and was able to finish the work by 1964, working under the direction of the NBS Chief of Numerical Analysis Philip J. Davis, who was also a contributor to the book. The major work of producing reliable mathematical tables, as described above, was part of the WPA project of Franklin Roosevelt. Legacy The Abramowitz Award is granted by the University of Maryland, College Park to students "for superior competence and promise in the field of mathematics and its applications." Winners of this award include Charles Fefferman and S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irene A
Irene is a name derived from εἰρήνη (eirēnē), the Greek for "peace". Irene, and related names, may refer to: * Irene (given name) Places * Irene, Gauteng, South Africa * Irene, South Dakota, United States * Irene, Texas, United States * Irene, West Virginia, United States * Irene Lake, Quebec, Canada * Lake Irene, a small lake in Rocky Mountain National Park, Colorado, United States * Lake Irene, a lake in Minnesota, United States * Irene River (Opawica River tributary), a tributary of the Opawica River in Quebec, Canada * Irene River (New Zealand), a river of New Zealand * Eirini metro station, an Athens metro station in Ano Maroussi, Greece Storms and hurricanes * Tropical Storm Irene (1947) * Tropical Storm Irene (1959) * Hurricane Irene–Olivia (1971) * Hurricane Irene (1981), part of the 1981 Atlantic hurricane season * Hurricane Irene (1999) * Hurricane Irene (2005) * Hurricane Irene (2011) Arts and entertainment Films and anime * ''Irene'' (1926 film), a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abramowitz And Stegun
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and Technology'' (NIST). Its full title is ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables''. A digital successor to the Handbook was released as the "Digital Library of Mathematical Functions" (DLMF) on 11 May 2010, along with a printed version, the ''NIST Handbook of Mathematical Functions'', published by Cambridge University Press. Overview Since it was first published in 1964, the 1046 page ''Handbook'' has been one of the most comprehensive sources of information on special functions, containing definitions, identities, approximations, plots, and tables of values of numerous functions used in virtually all fields of applied mathematics. The notation used in the ''Handbook'' is the '' de facto'' stan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
