The term "integral" can refer to a number of different concepts in mathematics. The most common meaning is the fundamental object of calculus corresponding to summing infinitesimal pieces to find the content of a continuous region. Other uses of "integral" include values that always take on integer values (e.g., integral embedding, integral graph), mathematical objects for which integers form basic examples (e.g., integral domain), and particular values of an equation (e.g., integral curve).
In calculus, 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 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.
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 
.
Many definitions of an integral are based on a particular measure. For instance, the Riemann integral is based on
Jordan measure, and the Lebesgue
integral is based on Lebesgue measure. Moreover,
depending on the context, any of a variety of other integral notations may be used.
For example, the Lebesgue integral of an integrable function 
over a set 
which is measurable
with respect to a measure 
is often written
 |
(3)
|
In the event that the set 
in (3) is an interval ![X=[a,b]](/images/equations/Integral/Inline11.svg)
, the "subscript-superscript"
notation from (2) is usually adopted. Another generalization
of the Riemann integral is the Stieltjes integral,
where a continuous integrand function 
on a closed interval ![I=[a,b]](/images/equations/Integral/Inline13.svg)
can be integrated against a real-valued function 
of bounded
variation defined on 
,
the result of which has the form
 |
(4)
|
or equivalently
 |
(5)
|
In differential geometry, form integration integrates a differential k-form 
over an oriented 
-dimensional submanifold or chain. In a positively oriented coordinate
chart, write 
.
For a set 
in this chart,
 |
(6)
|
where 
is 
-dimensional Lebesgue
measure. Reversing the orientation changes
the sign of the integral. Worth noting is that
the notation on the left-hand side of equation (6)
is similar to that in expression (5) above.
There are two classes of (Riemann) integrals: definite integrals such as (5), which have upper and lower limits, and indefinite integrals, such as
 |
(7)
|
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
 |
(8)
|
What's more, the first fundamental theorem of calculus can be rewritten more generally in terms of differential
forms (as in (6) above) to say that the integral
of a differential form 
over the boundary 
of some orientable
manifold 
equals the integral of the exterior
derivative 
over 
when 
is a compact, oriented smooth manifold and 
is a smooth form of degree 
, with the induced orientation
on the boundary, i.e.,
 |
(9)
|
Written in this form, the first fundamental theorem of calculus is known as Stokes' theorem.
Since the derivative of a constant is zero, indefinite integrals
are defined only up to an arbitrary constant
of integration 
,
i.e.,
 |
(10)
|
Wolfram Research maintains a web site https://www.wolframalpha.com/calculators/integral-calculator/ 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
 |
(11)
|
which is the first fundamental theorem of calculus. When 
is continuously differentiable
in its variables on a neighborhood of the integration
regions, and any moving limits are continuously
differentiable, other derivative-integral identities
include
 |
(12)
|
 |
(13)
|
(Kaplan 1992, p. 275), its generalization
 |
(14)
|
(Kaplan 1992, p. 258), and
![d/(dx)int_a^xf(x,t)dt=1/(x-a)int_a^x[(x-a)partial/(partialx)f(x,t)+(t-a)partial/(partialt)f(x,t)+f(x,t)]dt,](/images/equations/Integral/NumberedEquation15.svg) |
(15)
|
as follows from the moving-limit Leibniz integral rule and integration by parts, for 
.
Other integral identities include
 |
(16)
|
For a continuously differentiable vector field 
on a bounded region 
with piecewise smooth boundary,
the product rule and divergence
theorem give
 |
(17)
|
and
 |
(18)
|
where 
is the outward unit
normal vector. The boundary term vanishes, for example,
when 
on 
. Another integral identity is
 |
(19)
|
where 
is any function
and
 |
(20)
|
provided 
,
is real,
the defining series converges, and the integrals exist (Glasser 1983).
Integrals with rational exponents can often be evaluated by integration by
substitution using 
,
where 
is the least
common multiple of the denominators of the exponents.
