*For non-mathematical meanings of "Integral", see integration (non-mathematical).*

*It is recommended that the reader be familiar with algebra, derivatives, functions, and limitss.*

In mathematics, the term "**integral**" has two unrelated meanings; one relating to integers, the other relating to **integral calculus**.

Table of contents |

2 Integral Calculus 3 Improper and Trigonometric Integrals 4 Means of Integration 5 Riemann and Lebesgue Integrals 6 Other integrals |

## Integral Values

A real number is "**integral**" if it is an integer. The

**integral value**, of a real number

*x*, is defined as the largest integer which is less than, or equal to,

*x*; this is often denoted by ; known as the "floor function".

## Integral Calculus

In calculus, the **integral**, of a function, is the size of the area bounded by the x-axis and the graph of a function, *f*(*x*); negative areas are possible. Integrals are calculated by **integration**, which is a so-called "accumulation process" (see below).

Let *f*(*x*) be a function of the interval [*a*,*b*] into the real numbers. For simplicity, assume that this function is non-negative (it takes no negative values.) The set *S*=*S _{f}*:={(

*x*,

*y*)|0≤

*y*≤

*f*(

*x*)} is the region of the plane between

*f*and the

*x*axis. Measuring the "area" of

*S*is desirable, and this area is denoted by ∫

*f*, and it is the (definite) integral of

*f*.

## Improper and Trigonometric Integrals

If either the interval of integration, or the range of the function, is infinite; the integral is an "improper integral". Integrals which involve trigonometric functions, are trigonometric integrals. Some integrals can be evaluated via trigonometric substitution.## Means of Integration

The following pages discuss means of integrating various functions:- disk integration
- list of integrals
- shell integration
- trigonometric integration

## Riemann and Lebesgue Integrals

One should examine the articles on Riemann and Lebesgue integrals. The concept of Riemann integration was developed first, and Lebesgue integrals were developed to deal with pathological cases for which the Riemann integral was not defined. If a function is Riemann integrable, then it is also Lebesgue integrable, and the two integrals coincide.
The antiderivative approach occurs when we seek to find a function *F*(*x*) whose derivative *F*(*x*) is some given function *f*(*x*). This approach is motivated by calculus, and is the main method used for calculating the area under the curve as described in the preceding paragraph, for functions given by formulae.

Functions which have antiderivatives are also Riemann integrable (and hence Lebesgue integrable.) The nonobvious theorem that states that the two approaches ("area under the curve" and "antiderivative") are in some sense the same is the fundamental theorem of calculus

*(And the relationships works in reverse; the Radon-Nikodym derivative can be pulled out of the measure machinery underlying Lebesgue integrals.)*

### The nuance between Riemann and Lebesgue integration

Lastly, a limit-taking step is taken to make the elementary functions approach *f* more and more closely, and an area is obtained for some functions *f*. The functions which we can integrate are said to be *integrable*. However, the differences begin here; the Riemann theory was simpler thus far, but its simplicity results in a more limited set of integrable functions than the Lebesgue theory. In addition, the interaction between limits and the integral are more difficult to describe in the Riemann setting.

## Other integrals

- the Darboux integral, a variation of the Riemann integral
- the Denjoy integral, an extension of both the Riemann and Lebesgue integrals
- the Euler integral
- the Haar integral
- the Henstock-Kurzweil integral, an extension of both the Riemann and Lebesgue integrals (also called HK-integral)
- the Henstock-Kurzweil-Stieltjes integral (also called HK-Stieltjes integral)
- the Lebesgue-Stieltjes integral (also called Lebesgue-Radon integral)
- the Perron integral, which is equivalent to the restricted Denjoy integral
- the Stieltjes integral, an extension of the Riemann integral (also called Riemann-Stieltjes integral)

**See also**: Calculus, List of integrals