Fundamentals of integral calculus. Integral calculus

Integral calculus, a branch of mathematics that studies the properties and methods of calculating integrals and their applications. I. and. closely related to differential calculus and together with it constitutes one of the main parts of mathematical analysis (or infinitesimal analysis). The central concepts of I. and. are the concepts of a definite integral and an indefinite integral of functions of one real variable.

Definite integral. Suppose we need to calculate the area S“curvilinear trapezoid” - figures ABCD(cm. rice. ), bounded by an arc of a continuous line whose equation is at = f(x), segment AB x-axis and two ordinates AD And B.C. To calculate area S the base of this curved trapezoid AB(line segment [ a, b]) are divided into n sections (not necessarily equal) with dots A = x 0 < x 1 < ... < x n-1< < x n = b, denoting the lengths of these sections D x 1,D x 2, ..., D x n; on each such site, rectangles with heights are built f(x 1), f(x 2), ..., f(x n) where x k- some point from the segment [ x k - 1 , x k] (on rice. the rectangle constructed on the k-th section of the partition is shaded; f (x k) - its height). Sum S n the areas of the constructed rectangles are considered as an approximation to the area S curved trapezoid:

S» S n = f(x 1) D x 1 + f(x 2) D x 2 + f(x n) D x n

or, using the sum symbol S (the Greek letter sigma) to shorten the notation:

The indicated expression for the area of a curvilinear trapezoid is more accurate, the smaller the length D x k partition areas. To find the exact area value S need to find limit amounts S n under the assumption that the number of division points increases indefinitely and the largest of the lengths D x k tends to zero.

Abstracting from the geometric content of the problem considered, we come to the concept of a definite integral of the function f(x), continuous on the interval [ a, b], as to the limit of integral sums S n at the same limit. This integral is denoted

The symbol ò (extended S- the first letter of the word Summa) is called the integral sign, f(x) - integrand function, numbers A And b are called the lower and upper limits of the definite integral. If A = b, then, by definition, they assume

Besides,

Properties of the definite integral:

(k- constant). It is also obvious that

(the numerical value of a definite integral does not depend on the choice of notation for the integration variable).

The calculation of definite integrals reduces to problems of measuring areas bounded by curves (problems of “finding quadratures”), lengths of arcs of curves (“straightening curves”), surface areas of bodies, volumes of bodies (“finding cubatures”), as well as problems of determining the coordinates of centers of gravity , moments of inertia, the path of a body along a known speed of movement, the work produced by a force, and many other problems of natural science and technology. For example, the arc length of a plane curve given by the equation at = f(x) on the segment [ a, b], expressed by the integral

the volume of the body formed by the rotation of this arc around an axis Ox, - integral

the surface of this body - by the integral

The actual calculation of definite integrals is done in various ways. In some cases, a definite integral can be found by directly calculating the limit of the corresponding integral sum. However, for the most part, such a transition to the limit is difficult. Some definite integrals can be calculated by first finding indefinite integrals (see below). As a rule, one has to resort to an approximate calculation of definite integrals, using various quadrature formulas (For example, trapezoid formula , Simpson formula ). Such an approximate calculation can be carried out on a computer with an absolute error not exceeding any given small positive number. In cases that do not require great accuracy, graphical methods are used for approximate calculation of definite integrals (see. Graphics Computing ).

The concept of a definite integral extends to the case of an unbounded integration interval, as well as to some classes of unbounded functions. Such generalizations are called improper integrals .

Expressions like

where is the function f(x, a) is continuous in x are called parameter-dependent integrals. They serve as the main means of studying many special functions (see, for example, Gamma function ).

Indefinite integral. Finding indefinite integrals, or integration, is the inverse operation of differentiation. When differentiating a given function, its derivative is sought. When integrating, on the contrary, one looks for an antiderivative (or primitive) function - a function whose derivative is equal to the given function. So the function F(x) is an antiderivative for this function f(x), If F"(x) = f(x) or, what is the same, dF(x) = f(x) dx. This function f(x) may have different antiderivatives, but they all differ from each other only in their constant terms. Therefore, all antiderivatives for f(x) are contained in the expression F(x) + WITH, which is called the indefinite integral of the function f(x) and write down

Definite integral as a function of the upper limit of integration

(“integral with a variable upper limit”) is one of the antiderivatives of the integrand function. This allows you to establish the basic formula of I. and. (Newton-Leibniz formula):

expressing the numerical value of a certain integral in the form of the difference between the values of some antiderivative integrand function at the upper and lower limits of integration.

The mutually inverse nature of the operations of integration and differentiation is expressed by the equalities

This implies the possibility of obtaining from formulas and rules of differentiation the corresponding formulas and rules of integration (see table, where C, m, a, k- permanent and m¹ -1, A > 0).

Table of basic integrals and integration rules

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Difficulty I. and. in comparison with differential calculus is that integrals of elementary functions are not always expressed in terms of elementary functions; they may not be expressed, as they say, “in the final form.” I. and. has only separate methods of integration in its final form, the scope of each of which is limited (methods of integration are presented in textbooks of mathematical analysis: extensive tables of integrals are given in many reference books).

The class of functions whose integrals are always expressed in elementary functions includes the set of all rational functions

Where P(x) And Q(x) are polynomials. Many functions that are not rational also integrate in final form, for example functions that rationally depend on

or from x and rational powers of the fraction

In the final form, many transcendental functions are also integrated, for example, the rational functions of sine and cosine. Functions that are represented by indefinite integrals that are not taken in final form represent new transcendental functions. Many of them are well studied (see, for example, Integral logarithm , Integral sine and integral cosine , Integral exponential function ).

The concept of an integral extends to functions of many real variables (see Multiple integral , Curvilinear integral , Surface integral ), as well as functions of a complex variable (see. Analytical functions ) and vector functions (see Vector calculus ).

For expansion and generalization of the concept of integral, see Art. Integral.

Historical reference. The emergence of tasks of I. and. associated with finding areas and volumes. A number of problems of this kind were solved by mathematicians of Ancient Greece. Ancient mathematics anticipated the ideas of I. and. to a much greater extent than differential calculus. Played a major role in solving such problems exhaustion method , created Eudoxus of Cnidus and widely used Archimedes. However, Archimedes did not identify the general content of integration techniques and the concept of the integral, and even more so did not create an algorithm for artificial intelligence. Scientists of the Middle and Near East in the 9th-15th centuries. studied and translated the works of Archimedes into the Arabic language, which was generally available in their environment, but significantly new results in I. and. they didn't receive it. The activities of European scientists at this time were even more modest. Only in the 16th and 17th centuries. The development of natural sciences posed a number of new tasks for European mathematics, in particular the task of finding quadratures, cubatures and determining centers of gravity. The works of Archimedes, first published in 1544 (in Latin and Greek), began to attract widespread attention, and their study was one of the most important starting points for the further development of history. Antique "indivisible" method was revived I. Kepler. In a more general form, the ideas of this method were developed by B. Cavalieri , E. Torricelli , J. Wallis , B. Pascal. A number of geometric and mechanical problems were solved using the “indivisible” method. The later published works of P. date back to the same time. Farm by squaring parabolas n th degree, and then - the work of X. Huygens by straightening curves.

As a result of these studies, a commonality of integration techniques was revealed when solving seemingly dissimilar problems of geometry and mechanics, which were reduced to quadratures as the geometric equivalent of a definite integral. The final link in the chain of discoveries of this period was the establishment of mutual feedback between the problems of drawing tangents and quadratures, that is, between differentiation and integration. Basic concepts and algorithm of I. and. were created independently of each other. Newton and G. Leibniz. The latter belongs to the term “integral calculus” and the notation for the integral ò ydx.

Moreover, in Newton’s works the main role was played by the concept of the indefinite integral (fluents, see Fluxion calculus ), while Leibniz proceeded from the concept of a definite integral. Further development of I. and. in the 18th century associated with the names I. Bernoulli and especially L. Euler. At the beginning of the 19th century. I. and. O. was rebuilt together with differential calculus. Cauchy based on the theory of limits. In the development of I. and. in the 19th century Russian mathematicians M.V. took part. Ostrogradsky , V. Ya. Bunyakovsky , P.L. Chebyshev . At the end of the 19th - beginning of the 20th centuries. The development of set theory and the theory of functions of a real variable led to a deepening and generalization of the basic concepts of information and theory. (B. Riemann , A. Lebesgue and etc.).

Lit.: Story. Van der Waerden B. L., Awakening Science, trans. from Holland, M., 1959; Willeitner G., History of mathematics from Descartes to the mid-19th century, trans. from German, 2nd ed., M., 1966; Stroek D. Ya., Brief sketch of the history of mathematics, trans. from German, 2nd ed., M., 1969; Cantor M.. Vorleslingen ü ber Geschichte der Mathematik, 2 Aufl., Bd 3-4, Lpz. - B., 1901-24.

Works of the founders and classics of I. and. Newton I., Mathematical works, trans. from Latin, M.-L., 1937; Leibniz G., Selected excerpts from mathematical works, trans. With. Latin., “Advances in Mathematical Sciences”, 1948, vol. 3, century. 1; Euler L., Integral calculus, trans. from Latin, vol. 1-3, M., 1956-58; Koshy O. L., Summary of lessons on differential and integral calculus, trans. from French, St. Petersburg, 1831; his, Algebraic analysis, trans. from French, Leipzig, 1864.

Textbooks and teaching aids on I. and. Khinchin D. Ya., Short course in mathematical analysis, 3rd ed., 1957; Smirnov V.I., Course of Higher Mathematics, 22nd ed., vol. 1, M., 1967; Fikhtengolts G. M., Course of differential and integral calculus, 7th ed., vol. 2, M., 1969; Ilyin V., Poznyak E. G., Fundamentals of mathematical analysis, 3rd ed., part 1, M., 1971; Kurant R., Course of differential and integral calculus, trans. with him. and English, 4th ed., vol. 1, M., 1967; Dwight G.-B., Tables of integrals and other mathematical formulas, trans. from English, M., 1964.

Edited by Academician A. N. Kolmogorov.

Great Soviet Encyclopedia M.: "Soviet Encyclopedia", 1969-1978

Introduction

The integral symbol was introduced in 1675, and questions of integral calculus have been studied since 1696. Although the integral is studied mainly by mathematicians, physicists have also made their contribution to this science. Almost no physics formula can do without differential and integral calculus. Therefore, I decided to explore the integral and its application.

History of integral calculus

The history of the concept of integral is closely connected with problems of finding quadratures. Mathematicians of Ancient Greece and Rome called problems on the quadrature of a particular flat figure to calculate areas. The Latin word quadratura translates as “to square.” The need for a special term is explained by the fact that in ancient times (and later, up to the 18th century), ideas about real numbers were not yet sufficiently developed. Mathematicians operated with their geometric analogues, or scalar quantities, which cannot be multiplied. Therefore, problems for finding areas had to be formulated, for example, like this: “Construct a square equal in size to the given circle.” (This classical problem “on the squaring of a circle” cannot, as we know, be solved with the help of a compass and a ruler.)

The symbol t was introduced by Leibniz (1675). This sign is a modification of the Latin letter S (the first letter of the word summ a). The word integral itself was invented by J. Bernoulli (1690). It probably comes from the Latin integro, which translates as bringing to a previous state, restoring. (Indeed, the operation of integration “restores” the function by differentiating which the integrand was obtained.) Perhaps the origin of the term integral is different: the word integer means whole.

During the correspondence, I. Bernoulli and G. Leibniz agreed with J. Bernoulli’s proposal. At the same time, in 1696, the name of a new branch of mathematics appeared - integral calculus (calculus integralis), which was introduced by I. Bernoulli.

Other well-known terms related to integral calculus appeared much later. The name “primitive function”, now in use, has replaced the earlier “primitive function”, which was introduced by Lagrange (1797). The Latin word primitivus is translated as “initial”: F(x) = m f(x)dx - initial (or original, or antiderivative) for f (x), which is obtained from F(x) by differentiation.

In modern literature, the set of all antiderivatives for the function f(x) is also called the indefinite integral. This concept was highlighted by Leibniz, who noticed that all antiderivative functions differ by an arbitrary constant b, called a definite integral (the designation was introduced by C. Fourier (1768-1830), but Euler already indicated the limits of integration).

Many significant achievements of mathematicians of Ancient Greece in solving problems of finding quadratures (i.e. calculating areas) of plane figures, as well as cubatures (calculating volumes) of bodies are associated with the use of the exhaustion method proposed by Eudoxus of Cnidus (c. 408 - c. 355 BC .e.). Using this method, Eudoxus proved, for example, that the areas of two circles are related as the squares of their diameters, and the volume of a cone is equal to 1/3 of the volume of a cylinder having the same base and height.

Eudoxus' method was improved by Archimedes. The main stages characterizing Archimedes' method: 1) it is proved that the area of a circle is less than the area of any regular polygon described around it, but greater than the area of any inscribed; 2) it is proved that with an unlimited doubling of the number of sides, the difference in the areas of these polygons tends to zero; 3) to calculate the area of a circle, it remains to find the value to which the ratio of the area of a regular polygon tends when the number of its sides is unlimitedly doubled.

Using the exhaustion method and a number of other ingenious considerations (including the use of mechanics models), Archimedes solved many problems. He gave an estimate of the number p (3.10/71

Archimedes anticipated many of the ideas of integral calculus. (We add that in practice the first theorems on limits were proved by him.) But it took more than one and a half thousand years before these ideas found clear expression and were brought to the level of calculus.

Mathematicians of the 17th century, who obtained many new results, learned from the works of Archimedes. Another method was also actively used - the method of indivisibles, which also originated in Ancient Greece (it is associated primarily with the atomistic views of Democritus). For example, they imagined a curved trapezoid (Fig. 1, a) to be composed of vertical segments of length f(x), to which, nevertheless, they assigned an area equal to the infinitesimal value f(x)dx. In accordance with this understanding, the required area was considered equal to the sum

an infinitely large number of infinitely small areas. Sometimes it was even emphasized that the individual terms in this sum are zeros, but zeros of a special kind, which, added to an infinite number, give a well-defined positive sum.

On such a now seemingly at least dubious basis, J. Kepler (1571-1630) in his writings “New Astronomy”.

1609 and “Stereometry of Wine Barrels” (1615) correctly calculated a number of areas (for example, the area of a figure bounded by an ellipse) and volumes (the body was cut into 6 finitely thin plates). These studies were continued by the Italian mathematicians B. Cavalieri (1598-1647) and E. Torricelli (1608-1647). The principle formulated by B. Cavalieri, introduced by him under some additional assumptions, retains its significance in our time.

Let it be necessary to find the area of the figure shown in Figure 1, b, where the curves limiting the figure from above and below have the equations

y = f(x) and y=f(x)+c.

Imagining a figure made up of “indivisible”, in Cavalieri’s terminology, infinitely thin columns, we notice that they all have a total length c. By moving them in the vertical direction, we can form them into a rectangle with base b-a and height c. Therefore, the required area is equal to the area of the resulting rectangle, i.e.

S = S1 = c (b - a).

Cavalieri's general principle for the areas of plane figures is formulated as follows: Let the lines of a certain bundle of parallels intersect the figures Ф1 and Ф2 along segments of equal length (Fig. 1, c). Then the areas of the figures F1 and F2 are equal.

A similar principle operates in stereometry and is useful in finding volumes.

In the 17th century Many discoveries related to integral calculus were made. Thus, P. Fermat already in 1629 solved the problem of quadrature of any curve y = xn, where n is an integer (that is, he essentially derived the formula m xndx = (1/n+1)xn+1), and on this basis solved a series of problems to find centers of gravity. I. Kepler, when deducing his famous laws of planetary motion, actually relied on the idea of approximate integration. I. Barrow (1630-1677), Newton's teacher, came close to understanding the connection between integration and differentiation. Work on representing functions in the form of power series was of great importance.

However, despite the significance of the results obtained by many extremely inventive mathematicians of the 17th century, calculus did not yet exist. It was necessary to highlight the general ideas underlying the solution of many particular problems, as well as to establish a connection between the operations of differentiation and integration, which gives a fairly general algorithm. This was done by Newton and Leibniz, who independently discovered a fact known as the Newton-Leibniz formula. Thus, the general method was finally formed. He still had to learn how to find antiderivatives of many functions, give new logical calculus, etc. But the main thing has already been done: differential and integral calculus has been created.

Methods of mathematical analysis actively developed in the next century (first of all, the names of L. Euler, who completed a systematic study of the integration of elementary functions, and I. Bernoulli should be mentioned). Russian mathematicians M.V. took part in the development of integral calculus. Ostrogradsky (1801-1862), V.Ya. Bunyakovsky (1804-1889), P.L. Chebyshev (1821-1894). Of fundamental importance, in particular, were the results of Chebyshev, who proved that there are integrals that cannot be expressed through elementary functions.

A rigorous presentation of the integral theory appeared only in the last century. The solution to this problem is associated with the names of O. Cauchy, one of the greatest mathematicians, the German scientist B. Riemann (1826-1866), and the French mathematician G. Darboux (1842-1917).

Answers to many questions related to the existence of areas and volumes of figures were obtained with the creation of the theory of measure by C. Jordan (1838-1922).

Various generalizations of the concept of integral already at the beginning of our century were proposed by the French mathematicians A. Lebesgue (1875-1941) and A. Denjoy (188 4-1974), the Soviet mathematician A.Ya. Khinchinchin (1894-1959).

(287 BC - 212 BC): the essay “On the Measurement of the Circumference” discusses the issue of determining the area and circumference of a circle, and the treatise “On the Sphere and Cylinder” discusses surfaces and volumes of some bodies. To solve these problems, Archimedes used the exhaustion method of Eudoxus of Cnidus (c. 408 BC - c. 355 BC).

Thus, integral calculus arose from the need to create a general method for finding areas, volumes and centers of gravity.

These methods were systematically developed in the 17th century in the works of Cavalieri (1598-1647), Torricelli (1608-1647), P. Fermat (1601-1665), B. Pascal (1623-1662) and other scientists. But their research was mainly of a scattered and utilitarian nature - specific independent problems were solved. In 1659, I. Barrow (1630-1677) established a relationship between the problem of finding the area and the problem of finding the tangent.

The foundations of classical integral calculus were laid in the works of I. Newton (1643-1727) and G. Leibniz (1646-1716), who in the 70s of the 17th century abstracted from the mentioned particular applied problems and established a connection between integral and differential calculus. This allowed Newton, Leibniz and their students to develop the technique of integration. Integration methods mainly reached their current state in the works of L. Euler (1707-1783). The development of methods was completed by the works of M. V. Ostrogradsky (1801-1861) and P. L. Chebyshev (1821-1894).

Figure 1.1. Geometric interpretation of the Riemann integral.

Historically, the integral was understood as the area of a curvilinear trapezoid formed by a given curve and coordinate axis. To find this area, use a segment a b (\displaystyle ab) divided into n (\displaystyle n) not necessarily equal parts and built a stepped figure (it is shaded). Its area is equal

F n = y 0 d x 0 + y 1 d x 1 + … + y n − 1 d x n − 1 , (\displaystyle F_(n)=y_(0)\,dx_(0)+y_(1)\,dx_(1 )+\ldots +y_(n-1)\,dx_(n-1),)(1.1)

Where y i (\displaystyle y_(i))- function value f (x) (\displaystyle f(x)) V i (\displaystyle i)-that point ( i = 0 , 1 , … , n − 1 (\displaystyle i=0,\;1,\;\ldots ,\;n-1)), A d x i = x i + 1 − x i (\displaystyle dx_(i)=x_(i+1)-x_(i)).

G. Leibniz at the end of the 17th century designated the limit of this amount as

∫ y d x . (\displaystyle \int y\,dx.)(1.2)

At that time, the concept of a limit had not yet been formed, so Leibniz introduced a new symbol for the sum of an infinite number of terms ∫ (\displaystyle \int )- modified italic Latin “ ” - the first letter of Lat. summa(sum).

The word "integral" comes from the Latin. integralis- holistic. This name was proposed by Leibniz's student Johann Bernoulli (1667-1748) to distinguish the "sum of an infinite number of terms" from the ordinary sum.

Leibniz's notation was later improved by J. Fourier (1768-1830). He clearly began to indicate the start and end values x (\displaystyle x):

∫ a b y d x (\displaystyle \int \limits _(a)^(b)y\,dx)(1.3)

thereby introducing the modern designation definite integral.

In the theory of definite integrals, integration is considered as a process of generalizing summation to the case of an infinitely large number of infinitesimal expressions. Thus, the result of a certain integration (if possible) is a certain number (in generalizations, infinity).

Indefinite integral is a function (more precisely, a family of functions).

Integration, as opposed to differentiation, is considered an art, which is primarily due to the small number of laws that all integrals would satisfy. Moreover, for the existence of an integral, according to the main theorem of integral calculus, only the continuity of the integrable function is necessary. The fact of the existence of the integral does not provide at least some way of finding it in a closed form, that is, in the form of a finite number of operations on elementary functions. Much of the issue of finding integrals in closed form was solved in the works of J. Liouville (1809-1882). This topic was further developed in works devoted to the development of symbolic integration algorithms using a computer. An example is the Risch algorithm.

Wanting to emphasize the inverse nature of integration with respect to differentiation, some authors use the term “antidifferential” and denote the indefinite integral by the symbol D − 1 (\displaystyle D^(-1)).

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1. Antiderivative fufunction and indefinite integral

Integral calculus is the second part of the course in mathematical analysis, immediately following differential calculus. The very concept of an integral, along with the concept of derivative and differential, is a fundamental concept of mathematical analysis. This concept arose, on the one hand, from the need to solve problems of calculating area, circumference, volume, work of a variable force, center of gravity, etc., on the other hand, from the need to find functions from their derivatives.

In accordance with this, the concepts of definite and indefinite integrals arose.

As you know, the main task of differential calculus is to find the derivative or differential of a given function.

You can pose the inverse problem: given a function f(x), find a function F(x) that satisfies the condition F?(x)=f(x) or dF(x)=f(x)dx. Finding a function given its derivative or differential is one of the main tasks of integral calculus.

The problem of restoring a function from its derivative or differential leads to a wide variety of questions of mathematical analysis with its numerous applications in the fields of geometry, mechanics, physics, and technology.

Let us give an example; we encounter this kind of problem when, given the given speed of movement of a material point v=f(t), it is required to find the law of motion of this point, that is, the dependence of the path s traversed by the point on time t. In differential calculus we dealt with the inverse problem. There, according to the given law of motion s=s(t), by differentiating the function s(t), we found the speed v of this movement, that is, v(t)=s?(t). Therefore, in the problem posed above, we must, given the function v=f(t), reconstruct the function s=s(t), for which f(t) is the derivative.

Definition. The function F(x) is called an antiderivative function for the function f(x) on the interval X if at each point x of this interval F"(x)=f(x).

Thus, the function s(t) - a variable path - is an antiderivative for the speed v=f(t).

The function sin x is an antiderivative of the function cos x on the entire Ox axis, since for any value of x we will have: (sin x)?=cos x.

is an antiderivative of the function, since.

According to the geometric meaning of the derivative F"(x) is the angular coefficient of the tangent to the curve y=F(x) at the point with the abscissa x. Geometrically, finding an antiderivative for f(x) means finding a curve y=F(x) such that the angular the coefficient of the tangent to it at an arbitrary point x is equal to the value f(x) of the given function at this point (see Fig. 1.1).

For a given function f(x), its antiderivative is not uniquely defined. By differentiating, it is not difficult to verify that functions, and in general, where C is a certain number, are antiderivative for the function f(x) = x2. Similarly, in the general case, if F(x) is some antiderivative of f(x), then, since (Fx)+ C)"= F"(x)=f(x), functions of the form F(x)+ C , where C is an arbitrary number, are also antiderivatives for f(x).

Geometrically, this means that if one curve y = F (x) is found that satisfies the condition F "(x) = tan b = f (x), then by shifting it along the ordinate axis, we again obtain curves that satisfy the specified condition (since such a shift does not change the angular coefficient of the tangent at the point with the abscissa x) (see Fig. 1.1).

The question remains whether an expression of the form F(x)+C describes all antiderivatives for the function f(x). The answer to this is given by the following theorem.

Theorem. If F1 (x) and F2 (x) are antiderivatives for the function f (x) on some interval X, then there is a number C such that the equality

F2 (x)= F1 (x)+ C.

Since (F2(x)-F1(x)"=F"2 (x)-F" 1 (x)=f(x)-f(x)=0, then, by consequence of Lagrange’s theorem (see. § 8.1), there is a number C such that F2 (x) - F1 (x) = C or F2 (x) = F1 (x) + C

From this theorem it follows that if F(x) is an antiderivative for the function f(x), then an expression of the form F(x) + C, where C is an arbitrary number, specifies all possible antiderivatives for f(x).

Definition. The set of all antiderivatives for the function f(x) on the interval X is called the indefinite integral of the function f(x) and is denoted by f(x) dx, where is the sign of the integral, f(x) is the integrand, f(x)dx is integrand, and the variable x is the variable of integration.

So by definition,

f(x) dx=F(x)+C (1.1)

where F(x) is some antiderivative of f(x), C is an arbitrary constant.

Thus, the indefinite integral of any function is the general form of all antiderivatives for this function.

Formula (1.1) shows that if any antiderivative function for f(x) is known, then its indefinite integral is thereby known, and, therefore, the task of finding some specific antiderivative for f(x) is equivalent to the task of finding its indefinite integral .

In this regard, the question naturally arises: for every function f(x) defined on a certain interval, does there exist an antiderivative F(x) (and therefore an indefinite integral)? It turns out that it is not for everyone. However, if f(x) is continuous on some interval, then it has an antiderivative on it (and therefore an indefinite integral). In the case of a discontinuous function, we will only talk about integrating it in one of the continuity intervals.

For example, a function has a discontinuity only at x=0. Therefore, the intervals of continuity for it will be (0, +?) and (-?, 0). In the first of them, one of the antiderivatives for is ln(x). Hence,

However, for x from the interval (-?, 0) this formula is already meaningless (since ln(x) at x<0 не определён) . В этом случае одной из первообразных для будет уже не ln(x), а ln(-x), ибо

And it became,

Combining both cases, we arrive at the formula:

Restoring a function from its derivative, or, what is the same, finding the indefinite integral over a given integrand is called integration.

Since integration is the inverse action in relation to differentiation, therefore the verification of the correctness of the result of integration is carried out by differentiation of the sequential one: differentiation should give the integrand function.

Check that

Indeed, Therefore, the integral is taken correctly.

Let us now return to the mechanical problem posed at the beginning: to determine the distance traveled s at a given speed v=f(t). Since the speed of a moving point is the time derivative of the path, the problem comes down to finding an antiderivative for the function v=f(t) . Hence,

For definiteness, let us be given that the speed of movement of a point is proportional to time t, that is, v=at, where a is the proportionality coefficient. Then according to the formula we have:

Where C is an arbitrary constant. We have obtained countless solutions that differ from each other by a constant term. This uncertainty is explained by the fact that we did not record the moment in time t from which the traveled path s is measured. To obtain a completely definite solution to the problem, it is enough to know the value of s= at some initial time t= - these are the so-called initial values. Let, for example, we know that at the initial time t=0 the path is s=0. Then, assuming t=0, s=0 in the equality, we find 0=0+C, whence C=0. Consequently, the desired law of motion of a point is expressed by the formula.

Integral and the problem of determining area. Much more important is the interpretation of the antiderivative function as the area of a curvilinear figure. Since historically the concept of a antiderivative function was closely connected with the problem of determining area, we will dwell on this problem here.

Let a continuous function y=f(x) be given in the interval [a, b], taking only positive (non-negative) values. Consider the figure ABCD,

bounded by the curve y = f(x), two ordinates x = a and x = b and a segment of the x axis; We will call such a figure a curvilinear trapezoid. Wanting to determine the area P of this figure, we will study the behavior of the area of the variable figure AMND, contained between the initial ordinate x = a and the ordinate corresponding to a value x arbitrarily chosen in the interval. As x changes, this latter area will change accordingly, and each x corresponds to a completely definite value, so that the area of the curvilinear trapezoid AMND is a certain function of x; let us denote it by P(x).

Let us first set ourselves the task of finding the derivative of this function. For this purpose, let us give x some (say, positive) increment Dx; then the area P(x) will receive an increment of DR.

Let us denote by m and M, respectively, the smallest and largest values of the function f(x) in the interval [x, x + Dx] and compare the area of the DR with the areas of rectangles constructed on the basis of Dx and having heights m and M. Obviously, Dx<ДР<М Дх, откуда

If Dx>0, then, due to continuity, m and M will tend to f(x), and then

Thus, we arrive at the theorem (usually called the theorem of Newton and Leibniz a): the derivative of the variable area P(x) with respect to the finite abscissa x is equal to the finite ordinate y = f(x). In other words, the variable area P(x) is an antiderivative function for a given function y = f(x). Among other antiderivatives, this antiderivative is distinguished by the fact that it turns to 0 at x = a. Therefore, if any antiderivative F(x) for the function f(x) is known,

P(x) = F(x) + C,

then the constant C can be easily determined by setting here x = a

so C=-F(a).

Finally

In particular, to obtain the area P of the entire curvilinear trapezoid ABCD, you need to take x = b:

P = F(b) - F(a).

As an example, let us find the area P(x) of a figure bounded by the parabola y = ax2, the ordinate corresponding to the given abscissa x, and a segment of the x axis;

since the parabola intersects the x-axis at the origin, then the initial value of x here is 0. For the function f(x) = ax2 it is easy to find the antiderivative: F(x) = This function just turns to 0 at x = 0, so

In view of the connection that exists between the calculation of integrals and finding the areas of plane figures, i.e. their quadrature, it has become common to call the calculation of integrals itself quadrature.

To extend everything said above to the case of a function that also takes negative values, it is enough to agree to consider the areas of the parts of the figure located under the x axis to be negative.

Thus, no matter what the function f(x) is continuous in the interval [a, b], one can always imagine its antiderivative function in the form of a variable area of a figure limited by the graph of this function. However, this geometric illustration, of course, cannot be considered proof of the existence of an antiderivative, since the very concept of area has not yet been substantiated.

2. Properties of the indefinite integralla

1. The derivative of the indefinite integral is equal to the integrand, i.e.

Differentiating the left and right sides of equality (2.1), we obtain:

integral antiderivative function derivative

2. The differential of the indefinite integral is equal to the integrand: i.e. (2.2)

By definition of differential and property 1 we have

3. The indefinite integral of the differential of some function is equal to this function plus an arbitrary constant:

where C is an arbitrary number

Considering the function F(x) as an antiderivative for some function f(x), we can write

and based on (2.2) the differential of the indefinite integral f(x)dx=dF(x), whence

Comparing properties 2 and 3, we can say that the operations of finding an indefinite integral and differential are mutually inverse (the signs of d and cancel each other out, in the case of property 3, up to a constant term).

4. The constant factor can be taken out of the integral sign, i.e. if b=const?0 , then

where b is a certain number.

Let's find the derivative of the function:

(see property 1). By corollary to Lagrange's theorem, there is a number C such that g(x)=C and that means. Since the indefinite integral itself is found up to a constant term, the constant C can be omitted in the final representation of property 4.

5. The integral of the algebraic sum of two functions is equal to the same sum of integrals of these functions, i.e.

Indeed, let F(x) and G(x) be antiderivatives for the functions f(x) and g(x):

Then the functions F(x)±G(x) are antiderivatives of the function f(x)±g(x). Hence,

Property 5 is valid for any finite number of summand functions.

3. Table of basic integrals

Here is a table of the main integrals. The table of integrals follows directly from the definition of the indefinite integral and the table of derivatives.








	A<х<а, а>0

The integrals contained in this table are usually called tabular.

Since the indefinite integral does not depend on the choice of the integration variable, all table integrals take place for any variable.

The process of finding the antiderivative is reduced to transforming the integrand into tabular form.

The simplest integrals can be found by expanding the integrand into terms. Each integral includes a constant of integration, but all of them can be combined into one, so usually when integrating an algebraic sum of functions, only one constant of integration is written.

4 . Examples of finding integrals

There are entire classes of integrals, which, depending on constant factors or exponents, can be found using generalized integration formulas. Let's list some of them.

where P(x) is an integer polynomial with respect to x.

where n is any real number n? - 1; t = 1,2,3,...

9. If we designate

(n = 1,2, 3,...), then

12. (n=1,2,...);

13. (n=1,2,...);

1.1. Find the integrals:

a) Let’s imagine the integral as a sum of integrals and use tabular integrals

Examination:

i.e., the derivative is equal to the integrand.

b) Let's put the first factor in brackets and present the integral as the difference of two integrals

c) Let's make the following transformations

d) Subtract and add one to the numerator

e) Replace the roots with negative powers and represent the integral as the difference of two integrals

f) We assume that the trigonometric unit is a factor in the numerator

1 = sin2 x + cos2 x, then

1.2. Find the integrals:

a) Let's imagine 9 as 32 and use the table integral (14), where a = 3

b) Let us reduce the integrand to form and use the table integral (8)

c) Let's use the table integral (10)

d) Let’s combine the factors in the integrand and use the table integral (4)

e) Transform as follows

The integration method based on the application of properties 4 and 5 is called the expansion method. 1.3. Using the expansion method, find the integrals:

Solution. Finding each of the integrals begins with transforming the integrand. In problems a) and b) we will use the corresponding formulas for abbreviated multiplication and subsequent division of the numerator by the denominator:

(see table integrals (2) and (3)). We draw attention to the fact that at the end of the solution we write down one general constant C, without writing out the constants from the integration of individual terms. In the future, when writing, we will omit the constants from the integration of individual terms as long as the expression contains at least one indefinite integral. The final answer will then have one constant.

c) Transforming the integrand, we get

(see table integral (6)).

d) Isolating the whole part from the fraction, we get

(see table integral (9)).

Literature

1. Chernenko V. D. Higher mathematics in examples and problems: In 3 volumes: T. 1..-- St. Petersburg: Politekhnika, 2003.-- 703 e.: ill.

2. Kremer N.Sh. Higher mathematics for economists - M.: UNITI, 2004-471p.

3. Shipachev V.S. Higher mathematics. Textbook for universities.-4th ed. Ster.-M.: Higher school. 1998.-479 pp.: ill.

4. Fikhtengolts G. M. Course of differential and integral calculus: In 3 volumes: T. 2..-810 p.

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Integral calculus

a branch of mathematics that studies the properties and methods of calculating integrals and their applications. I. and. closely related to differential calculus (See Differential calculus) and together with it constitutes one of the main parts of mathematical analysis (or infinitesimal analysis). The central concepts of I. and. are the concepts of a definite integral and an indefinite integral of functions of one real variable.

Definite integral. Suppose we need to calculate the area S“curvilinear trapezoid” - figures ABCD(cm. rice. ), bounded by an arc of a continuous line whose equation is at = f(x), segment AB x-axis and two ordinates AD And B.C. To calculate area S the base of this curved trapezoid AB(line segment [ a, b]) are divided into n sections (not necessarily equal) with dots A = x 0 x 1 x n-1 x n = b, denoting the lengths of these sections Δ x 1 , Δ x 2 , ..., Δ x n; on each such site, rectangles with heights are built f(ξ 1), f(ξ 2), ..., f(ξ n) where ξ k- some point from the segment [ x k - 1 , x k] (on rice. the rectangle constructed on the k-th section of the partition is shaded; f (ξ k) - its height). Sum S n the areas of the constructed rectangles are considered as an approximation to the area S curved trapezoid:

S≈S n = f(ξ 1) Δ x 1 + f(ξ 2) Δ x 2 + f(ξ n) Δ x n

or, using the sum symbol Σ (Greek letter sigma) to shorten the notation:

The indicated expression for the area of a curvilinear trapezoid is more accurate, the shorter the length Δ x k partition areas. To find the exact area value S we need to find the Sum Limit S n under the assumption that the number of division points increases indefinitely and the largest of the lengths Δ x k tends to zero.

The symbol ∫ (extended S- the first letter of the word Summa) is called the integral sign, f(x) - integrand function, numbers A And b are called the lower and upper limits of the definite integral. If A = b, then, by definition, they assume

Properties of the definite integral:

(k- constant). It is also obvious that

the volume of the body formed by the rotation of this arc around an axis Ox, - integral

The actual calculation of definite integrals is done in various ways. In some cases, a definite integral can be found by directly calculating the limit of the corresponding integral sum. However, for the most part, such a transition to the limit is difficult. Some definite integrals can be calculated by first finding indefinite integrals (see below). As a rule, one has to resort to an approximate calculation of definite integrals, using various Quadrature formulas (for example, the trapezoidal formula (See Trapezoidal formula), Simpson's formula (See Simpson's formula)). Such an approximate calculation can be carried out on a computer with an absolute error not exceeding any given small positive number. In cases that do not require great accuracy, graphical methods are used for approximate calculation of definite integrals (see Graphical calculations).

The concept of a definite integral extends to the case of an unbounded integration interval, as well as to some classes of unbounded functions. Such generalizations are called improper integrals (See Improper integrals).

Expressions like

where is the function f(x, α) is continuous in x are called parameter-dependent integrals. They serve as the primary means of learning many special functions (See Special Functions) (see, for example, the Gamma function).

Definite integral as a function of the upper limit of integration

The mutually inverse nature of the operations of integration and differentiation is expressed by the equalities

This implies the possibility of obtaining from formulas and rules of differentiation the corresponding formulas and rules of integration (see table, where C, m, a, k- permanent and m≠ -1, A > 0).

Table of basic integrals and integration rules

The class of functions whose integrals are always expressed in elementary functions includes the set of all rational functions

Where P(x) And Q(x) are polynomials. Many functions that are not rational also integrate in final form, for example functions that rationally depend on

or from x and rational powers of the fraction

In the final form, many transcendental functions are also integrated, for example, the rational functions of sine and cosine. Functions that are represented by indefinite integrals that are not taken in final form represent new transcendental functions. Many of them are well studied (see, for example, Integral logarithm, Integral sine and integral cosine, Integral exponential function).

Historical reference. The emergence of tasks of I. and. associated with finding areas and volumes. A number of problems of this kind were solved by mathematicians of Ancient Greece. Ancient mathematics anticipated the ideas of I. and. to a much greater extent than differential calculus. A major role in solving such problems was played by the Exhaustion method, created by Eudoxus of Cnidus (See Eudoxus of Cnidus) and widely used by Archimedes. However, Archimedes did not identify the general content of integration techniques and the concept of the integral, and even more so did not create an algorithm for artificial intelligence. Scientists of the Middle and Near East in the 9th-15th centuries. studied and translated the works of Archimedes into the Arabic language, which was generally available in their environment, but significantly new results in I. and. they didn't receive it. The activities of European scientists at this time were even more modest. Only in the 16th and 17th centuries. The development of natural sciences posed a number of new tasks for European mathematics, in particular the task of finding quadratures, cubatures and determining centers of gravity. The works of Archimedes, first published in 1544 (in Latin and Greek), began to attract widespread attention, and their study was one of the most important starting points for the further development of history. The ancient “indivisible” method (See Indivisible method) was revived by J. Kepler. In a more general form, the ideas of this method were developed by B. Cavalieri, E. Torricelli, J. Wallis, B. Pascal (See Pascal). A number of geometric and mechanical problems were solved using the “indivisible” method. The later works of P. Fermat on squaring parabolas date back to the same time. n th degree, and then - the works of H. Huygens a by straightening curves.

As a result of these studies, a commonality of integration techniques was revealed when solving seemingly dissimilar problems of geometry and mechanics, which were reduced to quadratures as the geometric equivalent of a definite integral. The final link in the chain of discoveries of this period was the establishment of mutual feedback between the problems of drawing tangents and quadratures, that is, between differentiation and integration. Basic concepts and algorithm of I. and. were created independently of each other by I. Newton and G. Leibniz. The latter belongs to the term “integral calculus” and the notation for the integral ∫ ydx.

Moreover, in Newton’s works the main role was played by the concept of the indefinite integral (fluents, see Fluxian calculus), while Leibniz proceeded from the concept of the definite integral. Further development of I. and. in the 18th century associated with the names of I. Bernoulli and especially L. Euler . At the beginning of the 19th century. I. and. together with differential calculus, it was rebuilt by O. Cauchy on the basis of the theory of limits. In the development of I. and. in the 19th century Russian mathematicians M. V. Ostrogradsky, V. Ya. Bunyakovsky, P. L. Chebyshev took part. At the end of the 19th - beginning of the 20th centuries. The development of set theory and the theory of functions of a real variable led to a deepening and generalization of the basic concepts of information and theory. (B. Riemann, A. Lebesgue, etc.).

Lit.:Story. Van der Waerden B. L., Awakening Science, trans. from Holland, M., 1959; Willeitner G., History of mathematics from Descartes to the mid-19th century, trans. from German, 2nd ed., M., 1966; Stroek D. Ya., Brief sketch of the history of mathematics, trans. from German, 2nd ed., M., 1969; Cantor M.. Vorleslingen über Geschichte der Mathematik, 2 Aufl., Bd 3-4, Lpz. - B., 1901-24.

Edited by Academician A. N. Kolmogorov.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what “Integral calculus” is in other dictionaries:

Integral calculus- Integral calculus. Construction of integral sums for calculating the definite integral of a continuous function f(x), the graph of which is the MN curve. INTEGRAL CALCULUS, a branch of mathematics in which the properties and methods of calculation are studied... ... Illustrated Encyclopedic Dictionary

A branch of mathematics that studies the properties and methods of calculating integrals and their applications to solving various mathematical, physical and other problems. Integral calculus was proposed in a systematic form in the 17th century. I. Newton and G... Big Encyclopedic Dictionary

Department of higher mathematics, the study of actions opposite to differential calculation, namely, the determination of the relationship between several variable quantities using a given differential equation from them. Thus, there is... ... Dictionary of foreign words of the Russian language

INTEGRAL CALCULUS, see CALCULUS ... Scientific and technical encyclopedic dictionary