“Integral”的概念、定义、习题解题方法及翻译-数学辞典

单词

Integral

释义

Integral

An integral is a mathematical object which can be interpreted as an Area or a generalization of Area. Integrals, together with Derivatives, are the fundamental objects of Calculus. Other words for integral include Antiderivative and Primitive. TheRiemann Integral is the simplest integral definition and the only one usually encountered in elementaryCalculus. The Riemann Integral of the function over from to is written

(1)

Every definition of an integral is based on a particular Measure. For instance, the Riemann Integral isbased on Jordan Measure, and the Lebesgue Integral is based on Lebesgue Measure. The process ofcomputing an integral is called Integration (a more archaic term for Integration is Quadrature), andthe approximate computation of an integral is termed Numerical Integration.

There are two classes of (Riemann) integrals: Definite Integrals

(2)

which have upper and lower limits, and Indefinite Integrals, which are written withoutlimits. The first Fundamental Theorem of Calculus allowsDefinite Integrals to be computed in terms of Indefinite Integrals,since if

is the Indefinite Integral for

, then

(3)

Wolfram Researchmaintains a web site whichwill integrate many common (and not so common) functions. However, it cannot solve some simple integrals such as

(4)

(5)

where

is the Dilogarithm. Mathematica

3.0 (Wolfram Research, Champaign, IL) gives an incorrect answer of

(6)

although integrals of this type remain unevaluated in Mathematica 4.0. Integrals of this form

(7)

have a ``trick'' solution which takes advantage of the trigonometric identity

(8)

Letting




			(9)

Here is a list of common Indefinite Integrals:

			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)

			(27)
			(28)
			(29)
			(30)
			(31)
			(32)
			(33)
			(34)
			(35)
			(36)
			(37)
			(38)

where

is the Sine;

is the Cosine;

is the Tangent;

is theCosecant;

is the Secant;

is the Cotangent;

is the InverseCosine;

is the Inverse Sine;

is the Inverse Tangent;

, and

are Jacobi Elliptic Functions;

is a complete Elliptic Integral of the Second Kind; and

isthe Gudermannian Function.

To derive (15), let , so and



			(39)

To derive (16), let

, so

and




			(40)

To derive (19), let

(41)

(42)

and




			(43)

To derive (21), let

, so

and


			(44)

Differentiating integrals leads to some useful and powerful identities, for instance

(45)

which is the first Fundamental Theorem of Calculus. Other derivative-integralidentities include

(46)

the Leibniz Integral Rule

(47)

and its generalization

(48)

is singular or Infinite, then

(49)

Other integral identities include

(50)

(51)

(52)


			(53)

Integrals of the form

(54)

with one Infinite Limit and the other Nonzero may be expressed as finite integrals over transformed functions. If

decreases at least as fast as

, then let

			(55)
			(56)
			(57)

and

(58)

diverges as

for

, let

			(59)

			(60)
			(61)

and

(62)

diverges as

for

, let

			(63)
			(64)
			(65)

and

(66)

If the integral diverges exponentially, then let

			(67)
			(68)
			(69)

and

(70)

Integrals with rational exponents can often be solved by making the substitution , where is the Least Common Multiple of the Denominator of the exponents.

Integration rules include

(71)

(72)

For

(73)

is continuous on

and

is continuous and has an antiderivative on an Interval containing the values of

for

, then

(74)

Liouville showed that the integrals

(75)

cannot be expressed as terms of a finite number of elementary functions. Other irreducibles include

(76)

Chebyshev proved that if

, and

are Rational Numbers, then

(77)

is integrable in terms of elementary functions Iff

, or

is an Integer (Ritt 1948, Shanks 1993).

There are a wide range of methods available for Numerical Integration. A good source for such techniques is Press et al. (1992). The most straightforward numerical integration technique uses the Newton-Cotes Formulas (also calledQuadrature Formulas), which approximate a function tabulated at a sequence of regularly spacedIntervals by various degree Polynomials. If the endpoints are tabulated, then the 2-and 3-point formulas are called the Trapezoidal Rule and Simpson's Rule, respectively. The 5-point formula iscalled Bode's Rule. A generalization of the Trapezoidal Rule is Romberg Integration, which can yieldaccurate results for many fewer function evaluations.

If the analytic form of a function is known (instead of its values merely being tabulated at a fixed number of points), the bestnumerical method of integration is called Gaussian Quadrature. By picking the optimal Abscissas atwhich to compute the function, Gaussian quadrature produces the most accurate approximations possible. However, given the speedof modern computers, the additional complication of the Gaussian Quadrature formalism often makes it less desirable thanthe brute-force method of simply repeatedly calculating twice as many points on a regular grid until convergence is obtained. Anexcellent reference for Gaussian Quadrature is Hildebrand (1956).

See also A-Integrable, Abelian Integral, Calculus, Chebyshev-Gauss Quadrature, ChebyshevQuadrature, Darboux Integral, Definite Integral, Denjoy Integral, Derivative, DoubleExponential Integration, Euler Integral, Fundamental Theorem of Gaussian Quadrature, Gauss-JacobiMechanical Quadrature, Gaussian Quadrature, Haar Integral, Hermite-Gauss Quadrature, HermiteQuadrature, HK Integral, Indefinite Integral, Integration, Jacobi-Gauss Quadrature,Jacobi Quadrature, Laguerre-Gauss Quadrature, Laguerre Quadrature, Lebesgue Integral,Lebesgue-Stieltjes Integral, Legendre-Gauss Quadrature, Legendre Quadrature, LobattoQuadrature, Mechanical Quadrature, Mehler Quadrature, Newton-Cotes Formulas, NumericalIntegration, Peron Integral, Quadrature, Radau Quadrature, Recursive Monotone StableQuadrature, Riemann-Stieltjes Integral, Romberg Integration, Riemann Integral, StieltjesIntegral

References

Beyer, W. H. ``Integrals.'' CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 233-296, 1987.

Bronstein, M. Symbolic Integration I: Transcendental Functions. New York: Springer-Verlag, 1996.

Gordon, R. A. The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Providence, RI: Amer. Math. Soc., 1994.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1993.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319-323, 1956.

Piessens, R.; de Doncker, E.; Uberhuber, C. W.; and Kahaner, D. K. QUADPACK: A Subroutine Package for Automatic Integration. New York: Springer-Verlag, 1983.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Integration of Functions.'' Ch. 4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 123-158, 1992.

Ritt, J. F. Integration in Finite Terms. New York: Columbia University Press, p. 37, 1948.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 145, 1993.

Wolfram Research. ``The Integrator.'' http://www.integrals.com

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