This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.
is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.
The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals.
If the interval is infinite the definite integral is called an improper integral and defined by using appropriate limiting procedures. for example:
A constant, such pi, that may be defined by the integral of an algebraic function over an algebraic domain is known as a period.
The following is a list of some of the most common or interesting definite integrals. For a list of indefinite integrals see List of indefinite integrals.
and 29 Related for: List of definite integrals information
trigonometric functions, see Listofintegralsof exponential functions. For a complete listof antiderivative functions, see Lists ofintegrals. For the special...
multiple integralsof a single-variable function, see the Cauchy formula for repeated integration. Just as the definiteintegralof a positive function of one...
integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute the definiteintegral...
functions Common integrals in quantum field theory Normal distribution Listofintegralsof exponential functions Error function Berezin integral Stahl, Saul...
slightly different theorems called the second mean value theorem for definiteintegrals. A commonly found version is as follows: If G : [ a , b ] → R {\displaystyle...
Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definiteintegralof a function over a closed interval...
Dirichlet's test for improper integrals. It is a good illustration of special techniques for evaluating definiteintegrals, particularly when it is not...
(hx)}{1+x^{2}}}\,dx} is Owen's T function. Owen has an extensive listof Gaussian-type integrals; only a subset is given below. ∫ φ ( x ) d x = Φ ( x ) + C...
context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over which the integral is taken...
path integral formulation of quantum mechanics actually refers not to path integrals in this sense but to functional integrals, that is, integrals over...
numerical integration comprises a broad family of algorithms for calculating the numerical value of a definiteintegral. The term numerical quadrature (often abbreviated...
Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions.: 13–15 ...
Jørgen Pedersen Gram Skewes' number Listofintegralsof logarithmic functions Weisstein, Eric W. "Logarithmic Integral". MathWorld. Abramowitz and Stegun...
everywhere. The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock–Kurzweil integrals. Specifically, if a continuous...
The double integrals are surface integrals over the surface Σ, and the line integral is over the bounding curve ∂Σ. The Leibniz integral rule can be...
{1}{6}}(2x^{3}+1)^{7}(6x^{2})=(2x^{3}+1)^{7}(x^{2}).} For definiteintegrals, the limits of integration must also be adjusted, but the procedure is mostly...
definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definiteintegrals. It can also be interpreted...
the Riemann integrals for functions from R n → R {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} } are multiple integrals. The Riemann integral is unsuitable...
simplify the answer. Like other methods of integration by substitution, when evaluating a definiteintegral, it may be simpler to completely deduce the...
to indefinite integrals, while most of the integralsof interest to physicists, theoretical chemists, and engineers are definiteintegrals often related...
transcendental, special function of a single variable. It can variously be expressed in the form of a definiteintegral, a trigonometric series, and various...
{\displaystyle x=b} and applying the fundamental theorem of calculus gives the definiteintegral version: ∫ a b u ( x ) v ′ ( x ) d x = u ( b ) v ( b )...
useful for evaluating various trigonometric integrals. In particular, it can be used to evaluate the integralof the secant cubed, which, though seemingly...
predictions. INTEGRAL has far exceeded its 2+3-year planned lifetime, and is set to enter Earth atmosphere in 2029 as a definite end of the mission. Its...
functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential...
Gradshteyn and Ryzhik (GR) is the informal name of a comprehensive table ofintegrals originally compiled by the Russian mathematicians I. S. Gradshteyn...
Global Information
![{\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\left[\int _{a}^{b}f(x)\,dx\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a095fbd3f49faf63702fb3b314abd0e76c40f8a)