An integral is a mathematical object that generalizes the concept of area. Integrals and derivatives are fundamental objects of calculus. The Riemann integral is the simplest definition of an integral and is commonly used in physics and elementary calculus. An integral is computed by integrating a function over a range of values, and can be either definite or indefinite. The fundamental theorem of calculus relates definite integrals to indefinite integrals.

1. An integral is a mathematical object that 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 physics
and elementary calculus. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it
appears that cases where these methods [i.e., generalizations of the Riemann integral]
are applicable and Riemann's [definition of the integral] is not are too rare in physics to
repay 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, 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 such as (1), which have
upper and lower limits, and indefinite integrals, such as
(3)
which are written without limits. The first fundamental theorem of calculus allows definite
integrals to be computed in terms of indefinite integrals, since if is the indefinite
integral for , then
(4)
Since the derivative of a constant is zero, indefinite integrals are defined only up to an
arbitrary constant of integration , i.e.,
(5)
Wolfram Research maintains a web site http://integrals.wolfram.com/ that can find the

2. 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 identities
include
(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)

3. where is any function and
(16)
as long as and is real (Glasser 1983).