Editing Integral (section)

==Computation==

===Analytical===
The most basic technique for computing definite integrals of one real variable is based on the [[fundamental theorem of calculus]]. Let {{math|''f''(''x'')}} be the function of {{mvar|x}} to be integrated over a given interval {{math|[''a'', ''b'']}}. Then, find an antiderivative of {{mvar|f}}; that is, a function {{mvar|F}} such that {{math|''F''′ {{=}} ''f''}} on the interval. Provided the integrand and integral have no [[Mathematical singularity|singularities]] on the path of integration, by the fundamental theorem of calculus,

:<math>\int_a^b f(x)\,dx=F(b)-F(a).</math>

Sometimes it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include [[integration by substitution]], [[integration by parts]], [[Trigonometric substitution|integration by trigonometric substitution]], and [[Partial fractions in integration|integration by partial fractions]].

Alternative methods exist to compute more complex integrals. Many [[nonelementary integral]]s can be expanded in a [[Taylor series]] and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using [[Meijer G-function]]s can also be used, assuming that the integrand can be written as a product of Meijer G-functions. There are also many less common ways of calculating definite integrals; for instance, [[Parseval's identity]] can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see [[Gaussian integral]].

Computations of volumes of [[solid of revolution|solids of revolution]] can usually be done with [[disk integration]] or [[shell integration]].

Specific results which have been worked out by various techniques are collected in the [[Lists of integrals|list of integrals]].

===Symbolic===
{{Main|Symbolic integration}}

Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive [[Lists of integrals|tables of integrals]] have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to [[computer algebra system]]s that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration has been one of the motivations for the development of the first such systems, like [[Macsyma]] and [[Maple (software)|Maple]].

A major mathematical difficulty in symbolic integration is that in many cases, a relatively simple function does not have integrals that can be expressed in [[Closed-form expression|closed form]] involving only [[elementary function]]s, include [[rational function|rational]] and [[exponential function|exponential]] functions, [[logarithm]], [[trigonometric functions]] and [[inverse trigonometric functions]], and the operations of multiplication and composition. The [[Risch algorithm]] provides a general criterion to determine whether the antiderivative of an elementary function is elementary and to compute the integral if is elementary. However, functions with closed expressions of antiderivatives are the exception, and consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in [[Mathematica]], [[Maple (software)|Maple]] and other [[computer algebra system]]s, does just that for functions and antiderivatives built from rational functions, [[Nth root|radicals]], logarithm, and exponential functions.

Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the [[special functions]] (like the [[Legendre function]]s, the [[hypergeometric function]], the [[gamma function]], the [[incomplete gamma function]] and so on). Extending Risch's algorithm to include such functions is possible but challenging and has been an active research subject.

More recently a new approach has emerged, using [[D-finite function| ''D''-finite functions]], which are the solutions of [[linear differential equation]]s with polynomial coefficients. Most of the elementary and special functions are ''D''-finite, and the integral of a ''D''-finite function is also a ''D''-finite function. This provides an algorithm to express the antiderivative of a ''D''-finite function as the solution of a differential equation. This theory also allows one to compute the definite integral of a ''D''-function as the sum of a series given by the first coefficients and provides an algorithm to compute any coefficient.

Rule-based integration systems facilitate integration.  Rubi, a computer algebra system rule-based integrator, pattern matches an extensive system of symbolic integration rules to integrate a wide variety of integrands.  This system uses over 6600 integration rules to compute integrals.{{sfn|Rich|Scheibe|Abbasi|2018}}  The [[Ramanujan's master theorem#Bracket integration method| method of brackets]] is a generalization of Ramanujan's master theorem that can be applied to a wide range of univariate and multivariate integrals.  A set of rules are applied to the coefficients and exponential terms of the integrand's power series expansion to determine the integral.  The method is closely related to the [[Mellin transform]].{{sfn|Gonzalez|Jiu|Moll|2020}}

===Numerical===
{{Main|Numerical integration}}
[[File:Numerical_quadrature_4up.png|right|thumb|Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature]]
Definite integrals may be approximated using several methods of [[numerical integration]]. The [[rectangle method]] relies on dividing the region under the function into a series of rectangles corresponding to function values and multiplies by the step width to find the sum. A better approach, the [[trapezoidal rule]], replaces the rectangles used in a Riemann sum with trapezoids. The trapezoidal rule weights the first and last values by one half, then multiplies by the step width to obtain a better approximation.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=519–520}}.</ref> The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further: [[Simpson's rule]] approximates the integrand by a piecewise quadratic function.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=522–524}}.</ref>

Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called the [[Newton–Cotes formulas]]. The degree {{mvar|n}} Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree ''{{mvar|n}}'' polynomial. This polynomial is chosen to interpolate the values of the function on the interval.<ref>{{Harvnb|Kahaner|Moler|Nash|1989|p=144}}.</ref> Higher degree Newton–Cotes approximations can be more accurate, but they require more function evaluations, and they can suffer from numerical inaccuracy due to [[Runge's phenomenon]]. One solution to this problem is [[Clenshaw–Curtis quadrature]], in which the integrand is approximated by expanding it in terms of [[Chebyshev polynomials]].

[[Romberg's method]] halves the step widths incrementally, giving trapezoid approximations denoted by {{math|''T''(''h''<sub>0</sub>)}}, {{Math|''T''(''h''<sub>1</sub>)}}, and so on, where {{math|''h''<sub>''k''+1</sub>}} is half of {{math|''h''<sub>''k''</sub>}}. For each new step size, only half the new function values need to be computed; the others carry over from the previous size. It then [[Interpolation|interpolate]] a polynomial through the approximations, and extrapolate to {{math|''T''(0)}}. [[Gaussian quadrature]] evaluates the function at the roots of a set of [[orthogonal polynomials]].<ref>{{Harvnb|Kahaner|Moler|Nash|1989|p=147}}.</ref> An {{mvar|n}}-point Gaussian method is exact for polynomials of degree up to {{math|2''n'' − 1}}.

The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as [[Monte Carlo integration]].<ref>{{Harvnb|Kahaner|Moler|Nash|1989|pp=139–140}}.</ref>

===Mechanical===
The area of an arbitrary two-dimensional shape can be determined using a measuring instrument called [[planimeter]]. The volume of irregular objects can be measured with precision by the fluid [[displacement (fluid)|displaced]] as the object is submerged.

===Geometrical===
{{main|Quadrature (mathematics)}}
Area can sometimes be found via [[geometrical]] [[compass-and-straightedge construction]]s of an equivalent [[square]].

===Integration by differentiation===
Kempf, Jackson and Morales demonstrated mathematical relations that allow an integral to be calculated by means of [[Derivative|differentiation]]. Their calculus involves the [[Dirac delta function]] and the [[partial derivative]] operator <math>\partial_x</math>. This can also be applied to [[functional integral]]s, allowing them to be computed by [[functional derivative|functional differentiation]].<ref>{{Harvnb|Kempf|Jackson|Morales|2015}}.</ref>