An integral is a mathematical object that can be interpreted as an area or ageneralization of area. Integrals, together with derivatives, are the fundamental objects ofcalculus. Other words for integral include antiderivative and primitive. The Riemannintegral is the simplest integral definition and the only one usually encountered in physicsand elementary calculus. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "itappears that cases where these methods [i.e., generalizations of the Riemann integral]are applicable and Riemanns [definition of the integral] is not are too rare in physics torepay the extra difficulty."The Riemann integral of the function over from to is written (1)Note that if , the integral is written simply (2)as opposed to .Every definition of an integral is based on a particular measure. For instance, theRiemann integral is based on Jordan measure, and the Lebesgue integral is based onLebesgue measure. The process of computing an integral is called integration (a morearchaic term for integration is quadrature), and the approximate computation of anintegral is termed numerical integration.There are two classes of (Riemann) integrals: definite integrals such as (1), which haveupper and lower limits, and indefinite integrals, such as (3)which are written without limits. The first fundamental theorem of calculus allows definiteintegrals to be computed in terms of indefinite integrals, since if is the indefiniteintegral for , then (4)Since the derivative of a constant is zero, indefinite integrals are defined only up to anarbitrary constant of integration , i.e., (5)Wolfram Research maintains a web site http://integrals.wolfram.com/ that can find the
indefinite integral of many common (and not so common) functions.Differentiating integrals leads to some useful and powerful identities. For instance, if is continuous, then (6)which is the first fundamental theorem of calculus. Other derivative-integral identitiesinclude (7)theLeibniz integral rule (8)(Kaplan 1992, p. 275), its generalization (9)(Kaplan 1992, p. 258), and (10)as can be seen by applying (9) on the left side of (10) and using partial integration.Other integral identities include (11) (12) (1 3) (1 4)and the amusing integral identity (15)
where is any function and (16)as long as and is real (Glasser 1983).