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. The
Riemann Integral is the simplest integral definition and the only one usually encountered in elementary
Calculus. 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 is based on Jordan Measure, and the Lebesgue Integral is based on Lebesgue Measure. The process of computing an integral is called Integration (a more archaic term for Integration is Quadrature), and the approximate computation of an integral is termed Numerical Integration.

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

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(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 the Cosecant; is the Secant; is the Cotangent; is the Inverse Cosine; is the Inverse Sine; is the Inverse Tangent; , , and are Jacobi Elliptic Functions; is a complete Elliptic Integral of the Second Kind; and is the Gudermannian Function.

To derive (15), let
, so
and

(39) |

To derive (16), let , so and

(40) |

To derive (19), let

(41) |

(42) |

(43) |

To derive (21), let , so and

(44) |

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

(45) |

(46) |

(47) |

(48) |

(49) |

Other integral identities include

(50) |

(51) |

(52) |

(53) |

Integrals of the form

(54) |

(55) | |||

(56) | |||

(57) |

and

(58) |

(59) | |||

(60) | |||

(61) |

and

(62) |

(63) | |||

(64) | |||

(65) |

and

(66) |

(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) |

(73) |

(74) |

Liouville showed that the integrals

(75) |

(76) |

(77) |

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 called
Quadrature Formulas), which approximate a function tabulated at a sequence of regularly spaced
Intervals 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 is
called Bode's Rule. A generalization of the Trapezoidal Rule is Romberg Integration, which can yield
accurate 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 best numerical method of integration is called Gaussian Quadrature. By picking the optimal Abscissas at which to compute the function, Gaussian quadrature produces the most accurate approximations possible. However, given the speed of modern computers, the additional complication of the Gaussian Quadrature formalism often makes it less desirable than the brute-force method of simply repeatedly calculating twice as many points on a regular grid until convergence is obtained. An excellent reference for Gaussian Quadrature is Hildebrand (1956).

**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

© 1996-9

1999-05-26