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Prudnikov, Brychkov And Marichev
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., an integral assigns numbers to functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, 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 int ... [...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: \ln x, \ \log_a x ... [...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 ... [...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\mbo ... [...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 integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity. Integrals of polynomials : \int xe^\,dx = e^\left(\frac\right) \text c \neq 0; : \int x^2 e^\,dx = e^\left(\frac-\frac+\frac\right) : \begin \int x^n e^\,dx &= \frac x^n e^ - \frac\int x^ e^ \,dx \\ &= \left( \frac \right)^n \frac \\ &= e^\sum_^n (-1)^i\fracx^ \\ &= e^\sum_^n (-1)^\fracx^i \end : \int\frac\,dx = \ln, x, +\sum_^\infty\frac : \int\frac\,dx = \frac\left(-\frac+c\int\frac\,dx\right) \qquad\textn\neq 1\text Integrals involving only exponential functions : \int f'(x)e^\,dx = e^ : \int e^\,dx = \frac e^ : \int a^\,dx = \frac a^\qquad\texta > 0,\ a \ne 1 Integrals involving the error function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Inverse Hyperbolic Functions
The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals. * In all formulas the constant is assumed to be nonzero, and denotes the constant of integration. * For each inverse hyperbolic integration formula below there is a corresponding formula in the list of integrals of inverse trigonometric functions The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions. For a complete list of integral formulas, see lists of integrals. * The inverse trigonometric functions are also known .... Inverse hyperbolic sine integration formulas \int\operatorname(ax)\,dx= x\operatorname(ax)-\frac+C \int x\operatorname(ax)\,dx= \frac+ \frac- \frac+C \int x^2\operatorname(ax)\,dx= \frac- \frac+C \int x^m\operatorname(ax)\,dx= \frac- \frac\int\frac\,dx\quad(m\ne-1) \int\operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Hyperbolic Functions
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant ''a'' is assumed to be nonzero, and ''C'' denotes the constant of integration. Integrals involving only hyperbolic sine functions \int\sinh ax\,dx = \frac\cosh ax+C \int\sinh^2 ax\,dx = \frac\sinh 2ax - \frac+C \int\sinh^n ax\,dx = \frac(\sinh^ ax)(\cosh ax) - \frac\int\sinh^ ax\,dx \qquad\mboxn>0\mbox : also: \int\sinh^n ax\,dx = \frac(\sinh^ ax)(\cosh ax) - \frac\int\sinh^ax\,dx \qquad\mboxn0\mbox : also: \int\cosh^n ax\,dx = -\frac(\sinh ax)(\cosh^ ax) + \frac\int\cosh^ax\,dx \qquad\mboxn<0\mboxn\neq -1\mbox : also: |
List Of Integrals Of Inverse Trigonometric Functions
The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions. For a complete list of integral formulas, see lists of integrals. * The inverse trigonometric functions are also known as the "arc functions". * ''C'' is used for the arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives. * There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as ''sin−1'', ''asin'', or, as is used on this page, ''arcsin''. * For each inverse trigonometric integration formula below there is a corresponding formula in the list of integrals of inverse hyperbolic functions The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Trigonometric Functions
The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral. Generally, if the function \sin x is any trigonometric function, and \cos x is its derivative, : \int a\cos nx\,dx = \frac\sin nx+C In all formulas the constant ''a'' is assumed to be nonzero, and ''C'' denotes the constant of integration. Integrands involving only sine : \int\sin ax\,dx = -\frac\cos ax+C : \int\sin^2 \,dx = \frac - \frac \sin 2ax +C= \frac - \frac \sin ax\cos ax +C : \int\sin^3 \,dx = \frac - \frac +C : \int x\sin^2 \,dx = \frac - \frac \sin 2ax - \frac \cos 2ax +C : \int x^2\sin^2 \,dx = \frac - \left( \frac - \frac \right) \sin 2ax - \frac \cos 2ax +C :\int x\sin ax\,dx = \fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Irrational Functions
The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the 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 ... is omitted for brevity. Integrals involving ''r'' = : \int r\,dx = \frac\left(x r +a^2\,\ln\left(x+r\right)\right) : \int r^3\,dx = \fracxr^3+\fraca^2xr+\fraca^4\ln\left(x+r\right) : \int r^5\,dx = \fracxr^5+\fraca^2xr^3+\fraca^4xr+\fraca^6\ln\left(x+r\right) : \int x r\,dx = \frac : \int x r^3\,dx = \frac : \int x r^\,dx = \frac : \int x^2 r\,dx = \frac-\frac-\frac\ln\left(x+r\right) : \int x^2 r^3\,dx = \frac-\frac-\frac-\frac\ln\left(x+r\right) : \int x^3 r\,dx = \frac - \frac : \int x^3 r^3\,dx = \frac-\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Rational Functions
The following is a list of integrals (antiderivative functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form: : \frac, and \frac. which can then be integrated term by term. For other types of functions, see lists of integrals. Miscellaneous integrands :\int\frac \, dx= \ln\left, f(x)\ + C :\int\frac \, dx = \frac\arctan\frac\,\! + C :\int\frac \, dx = \frac\ln\left, \frac\ + C = \begin \displaystyle -\frac\,\operatorname\frac + C = \frac\ln\frac + C & \text, x, , a, \mbox \end :\int\frac \, dx = \frac\ln\left, \frac\ + C = \begin \displaystyle \frac\,\operatorname\frac + C = \frac\ln\frac + C & \text, x, , a, \mbox \end : \int \frac = \frac\sum_^ \sin \left(\frac\pi\right) \arctan\left left(x - \cos \left(\frac\pi \right) \right ) \csc \left(\frac\pi \right) \right- \frac \cos \left(\frac\pi \right) \ln \left , x^2 - 2 x \cos \left(\frac\pi \right) + 1 \right , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
