The problem of integrating a function or computing the value of an *integral* function arises in the solution of practical problems such as the problem of acceleration of a particle under the action of an applied variable force, or the problem of finding the area of a curvilinear trapezoid bounded by the graph of a given function and coordinate segments. The first problem, formulated in the general form, introduces the notion of an indefinite integral; the second, the notion of a definite integral.

Finding the indefinite integral of a function f(x) is the operational inverse of differentiation where an antiderivative function F(x) is sought, such that its derivative is equal to f(x), i.e., F'(x) = f(x) and dF(x) = f(x)dx.

Hence, all the antiderivative functions of f(x) or its integrals can be expressed as

where c is an arbitrary constant.

The antiderivatives of elementary functions can easily be found on the basis of formulas for derivatives of these functions. The most widely used are:

A wide range of integrals is given in standard texts and handbooks.

Some other formulas can be obtained using the rules of integration, the simplest of which considers the property of operational linearity

In addition, methods of integration—such as a change of variables or integration by parts—are extensively used. The basis of the method of change of variables or substitution is the transformation of the independent variable. If

then the integral

can be easily calculated by changing the variable to t = w(x) according to

For the method of integration by parts, the rule of differentiation of product functions applies. Let u = f(x), v = g(x) be functions with continuous derivatives. Then from d(u v) = u dv + v dv, it follows that

Indefinite integrals of many even comparatively simple functions cannot be expressed in terms of elementary integrable functions by algebraic manipulation. Examples of this type include

Functions whose indefinite integrals cannot be obtained by simple methods are called *transcendental functions*. They are, however, widely used in analysis. Among these are the sine and cosine integral functions, integral exponential function, probability integral, Fresnel's integrals, Gamma Functions, Bessel Functions, etc. Numerical and other techniques can be used to obtain values for such integrals and tables of values are given in standard texts.

The notion of a definite integral, on the other hand, brings about the problem of solving the area of a curvilinear trapezoid bounded by a graph in the *Cartesian coordinate* system of functions f(x) > 0 in the interval [a, b], which is the base of a trapezoid and the vertical intervals [a, f(a)] and [b, f(b)].

To calculate this area, the trapezoid is divided into n rectangles whose bases are the intervals [x_{i}, x_{i+1}] of an arbitrary partitioning of the base [a, b] into n parts according to

and the heights are f(ξ_{i}), where ξ_{i} is an arbitrary point on the interval [x_{i}, x_{i+1}], i = 0, 1, ..., n−1.

The sum of the areas of rectangles

yields an approximate value of the area of a trapezoid. The exact value of the area can be obtained as a limit

for n → ∞, with the maximum interval of partitioning tending towards zero.

Leaving aside the problem of area definition, a finite limit of the sum S_{n} is called the *definite integral* of the function f(x) over the interval [a, b] and is denoted by

If this limit exists, the function f(x) is called an integrable function in the interval [a, b]. The numbers a and b are called the lower and upper limits of integration, the sum S_{n} is called the integral sum.

Functions integrable over an interval [a, b] include continuous functions, bounded functions having only finite number of discontinuity points, and monotonic functions.

If the two functions f(x) and g(x) are integrable over the interval [a, b], then their sum, difference and products are also integrable. If a function is integrable over [a, b], then it is integrable on any of its parts [α, β] [a, b].

It follows from the definition that

It has been proven that if f(x) and g(x) are integrable on given intervals, then the following properties of definite integral are also valid:

; where m ≤ μ ≤ M (the generalized mean-value theorem);

if f(x) decreases monotonically in the range [a, b] and f(x) ≥ 0, then , where ξ is a value from the interval [a, b]. if f(x) ≥ 0 and increases monotonically, then , (the second mean-value theorem);

, where uv = u(b)v(b) − u(a)v(a) (integration by parts formula);

if the product of a continuous function and its first derivative g(t) g'(t) [a, b] for t [α, β]. g(α) = a, g(β) = b, then

If a function f(x) is integrable over [a, b], then it is also integrable over [a, x] where x is any value in the range [a, b]. The function

will be continuous over [a, b], and at points where f(x) is continuous, Φ'(x) = f(x).

Thus, Φ(x) is an example of an antiderivative function f(x); any antiderivative function has the form

or

Having defined the constant c as

yields

and finally, the main formula of integral calculus, the *Newton-Leibnitz formula*

Where the integrand depends on several variables, the integrals which depend on the parameter are considered. These integrals are of the form

Subject to some limitations on f, such integrals can be differentiated and integrated "under the integral", i.e.,

In cases of integrals with variable limits of integration a(x) and b(x), the differentiation formula has the form

The notion of the definite integral is generalized to the case of an infinite interval of integration and also to some classes of unbounded functions, thus leading to improper integrals. Therefore, if the function f(x) is continuous on the interval [a, ∞] and there exists a limit

then it is called the integral with an infinite upper limit and is denoted by

If, in the interval [a, b], the function f(x) has one point of discontinuity C in which it is unbounded (otherwise, the definite integral exists in the previous sense), then the improper integral means

if these limits exist.

The notion of integrals extends from the functions of a complex variable to vector-functions of many real variables (multiple integrals).

Note the integral relations for the functions of many variables used in the mechanics of continuous medium:

*Green's formula*:

where L is the boundary of a plane area, and integration with respect to L is performed in a direction such that the area D is on the left;

*Gauss'-Ostrogradsky 's formula*:

where the integral on the right-hand side is taken over the external side of surface S of volume V;

*Stokes formula*:

which is the generalization of Green's formula for the case of an arbitrary surface A, bounded by a closed contour L.

Green's, Gauss'-Ostrogradsky's and Stokes formulas are widely employed in formulating problems and theorems of hydrodynamics and convective heat- and mass transfer, and also in approximating the corresponding transfer equations in a controlled volume for numerical solution.

*Stieltjes* and *Lebesque integrals* are further generalizations of the notion of integrals.

The Stieltjes integral of a function f(x), continuous on the interval [a, b], with respect to a bounded monotonic function u(x) is defined as the following sum:

where ξ_{i} are arbitrary points placed at intervals [x_{i}; x_{i+1}] in an arbitrary partitioning of the interval [a, b], for Δx_{i} = x_{i+1} − x_{i} → 0. This limit is denoted as:

The Stieltjes integral also exists when the function u(x) can be represented in the form of a sum of two bounded monotonic functions:

If the function u(x) has a derivative u'(x) bounded in the range [a, b] and integrable in the ordinary sense (in the sense of Riemann), then Stieltjes integral is reduced to Riemann's integral

In order to determine Lebesque's integral of some function f(x) over the interval [a, b], the range of values of a variable y = f(x) is divided into small intervals (as distinct from Riemann's integral where the field x is divided)

and an integral sum of the form

is completed, where M_{i} is the set of points x in the interval [a, b], for which y_{i−1} ≤ f(x) < y_{i} and μ(M_{i}) are values of the set M_{i}, y_{i−1} ≤ η_{i} ≤ y_{i}. The function f(x) is called integrable in the Lebesque sense if the series converges absolutely for Δy_{i} = y_{i} − y_{i−1} → 0. The limit of these sums is called the Lebesque integral and is denoted by .

In heat and mass transfer theory dealing with sufficiently smooth functions, the Lebesque integral is practically never used except for some statistical models of poorly determined phenomena, such as turbulent fluctuations of the parameters of a flow.