Main theorem of analysis

Main theorem of analysis or Newton-Leibniz formula gives the relation between two operations: taking a definite integral and calculating the antiderivative

Wording

Consider the integral of the function y = f(x) within a constant number a up to the number x, which we will consider variable. We write the integral in the following form:

This type of integral is called an integral with a variable upper limit. Using the mean-in-definite integral theorem, it is easy to show that a given function is continuous and differentiable. And also the derivative of this function at the point x is equal to the integrable function itself. From here it follows that any continuous function has an antiderivative in the form of a quadrature: . And since the class of antiderivatives of the function f differs by a constant, it is easy to show that: the definite integral of the function f is equal to the difference between the values ​​of the antiderivatives at the points b and a

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Newton-Leibniz formula

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Once, my father and I were driving far away in a car. And this is a good reason for a smart conversation.

We are talking about the "basic theorems". The basic theorem of arithmetic is that any integer can be decomposed into a product of prime numbers, and in a unique way. The basic theorem of algebra is that a polynomial has as many roots as its degree (although there is hell with the formulations). And then the main theorem of analysis somehow flew out of my head then.

Father suggested that the fundamental theorem of analysis is the Newton-Leibniz theorem. “What is this about?” I asked. Father: “I don’t remember the exact wording, but something about the fact that integration is an operation inverse to differentiation.”

Wait, isn't that by definition?

As always with these fundamental theorems, what they say seems obvious after you've already gone through it. But in fact, it is the main theorem that allows us to consider integration and differentiation as inverse operations. Deeply anti-scientific reasoning will go further, where any mathematician will find 100500 formal errors, but this is not important now.

What is differentiation? This is when we draw a tangent at each point of the function and find the tangent of the angle at which it passes to the horizon, like this:

Now, if each point is assigned the found tangent, then a new function will be obtained, which is called the derivative. Let me remind you that the number e that the derivative of the function ex is equal to ex, that is, at each point, the tangent of the angle is just equal to the value of the function itself.

What is integration? This is finding the area of ​​\u200b\u200ba figure under the curve of a function bounded by some vertical boundaries a and b and the horizontal axis:

If you divide by an increasing number of rectangles and look at the limit of the sum of the areas, then you get just the area of ​​\u200b\u200bthis figure. This area is called the definite integral of the function y = f(x) on the segment [ a; b] and is marked like this:

Frankly, it is not at all obvious that the bullshit about angles and bullshit about the area are generally somehow connected.

And this is how they are connected. The inverse derivative of a function is called antiderivative. Antiderivative from f(x) is such a function g(x) that its derivative g´(x) = f(x). For example, the function y = x 2 + 8 derivative y = 2x. So for the function y = x function y = (x 2 / 2) + 4 is antiderivative.

It is easy to see that there are an infinite number of such functions. For example, the derivative of the function y = x 2 + 28 is also y = 2x. So for the function y = x function ( x 2 / 2) + 14 is also an antiderivative. This is logical, because the derivative is the angle at each point, and it is natural that it does not change depending on the height to which we vertically raise the entire graph of the function as a whole. So for the function x primitive is x 2 / 2 plus as much as you like.

So, it turns out, to find the area of ​​the figure under the function y = f(x) ranging from a before b, you need to take the values ​​of any of its antiderivatives g(x) at points b and a and subtract one from the other:

Here g- although any, but still some kind of one primitive, therefore, “as many as you like” will be the same for it, they will be subtracted from each other and will not affect the result. You can take some simple function like y = 2x, where the area without integrals is easy to calculate in your mind and check. Works!

This formula is called the fundamental theorem of analysis or the Newton-Leibniz theorem. If it is proved, then we can already call the finding of the antiderivative integration and generally treat differentiation and integration as mutually inverse operations.

§ 5. Main theorem of analysis

1. Main theorem. The concept of integration, and to some extent of differentiation, was well developed before the work of Newton and Leibniz. But it was absolutely necessary to make one very simple discovery in order to give impetus to the enormous evolution of the newly created mathematical analysis. Two apparently mutually non-contiguous limit processes, used one for differentiation, the other for integrating functions, turned out to be closely connected with each other. Indeed, they are mutual

				reverse operations,
				good for operations like
				addition and subtraction, smart
				cutting and division. Differen-
				social and integral
				numbers are
				something unified.
				The great achievement of New
				tone and Leibniz is
				in that for the first time they
Rice. 274. Int played as a function top				but understood and used
				this main theorem of analysis
				behind. Without a doubt, they are open

tie lay n but the direct path is scientific development, and not at all surprising Remarkably, the difference These individuals came independently and almost simultaneously to a clear understanding of the above circumstance.

In order to formulate the main theorem precisely, we consider the integral of the function y = f(x) in the range from a constant number a to a number x, which we will consider variable. In order not to confuse the upper limit of integration x with the variable appearing under the integral sign, we write the integral in the following form (see page 428):

F(x)=Z

thus demonstrating our intention to study the integral as a function F(x) of its upper limit (Fig. 274). This function F (x) is the area under the curve y = f(u) from the point u = a to the point u = x. Sometimes the integral F(x) with a variable upper limit is called the "indefinite integral".

The main theorem of analysis reads as follows:

The derivative of the indefinite integral (1) with respect to its upper limit x is equal to the value of the function f(u) at the point u = x:

F 0 (x) = f(x).

MAIN THEOREM OF ANALYSIS

In other words, the integration process leading from the function f(x) to the function F(x) is "destroyed" by the inverse process of differentiation applied to the function F(x).

On an intuitive basis, the proof of this proposition is not difficult. It is based on the interpretation of the integral F(x) as an area, and would be obscured if we tried to plot the function F(x) and interpret the derivative F0(x) as the corresponding slope. Leaving aside the previously established geometric interpretation of the derivative, we will retain the geometric interpretation of the integral F (x) as an area, and we will become an analytical method to differentiate the function F (x). Difference

F (x1 ) − F (x)

is simply the area under the curve y = f(u) between the limits u = x1 and u = x (Fig. 275), and it is easy to understand that the numerical value of this area lies between the numbers (x1 − x)m and (x1 − x) M:

(x1 − x)m 6 F (x1 ) − F (x) 6 (x1 − x)M,

where M and m are, respectively, the largest and smallest values ​​of the function f(u) in the interval from u = x to u = x1 . Indeed, these products give the areas of two rectangles, of which one contains the curvilinear region under consideration, and the other is contained in it.

Rice. 275. On the proof of the main theorem

this implies

m 6 F (x1 ) − F (x) 6 M. x1 − x

Let us assume that the function f(u) is continuous, so that as x1 tends to x, both quantities M and m tend to the value of the function f(u) at the point u = x, i.e., to the value of f(x). In this case, one can consider

468 MATHEMATICAL ANALYSIS Ch. VIII

proven that
F 0 (x) = lim	F (x1 ) − F (x)
x1→x	x1 − x

The intuitive meaning of this result is that as it increases, the rate of change of the area under the curve y = f(x) is equal to the height of the curve at x.

In some manuals, the content of this main theorem is obscured due to poorly chosen terminology. Namely, many authors first introduce the concept of a derivative, and then define the "indefinite integral" simply as the result of the inverse operation with respect to differentiation: they say that the function G(x) is an indefinite integral of the function f(x) if

G0 (x) = f(x).

Thus, this way of presentation directly connects differentiation with the word "integral". It is only later that the concept of "definite integral" is introduced, treated as an area or as the limit of a sequence of sums, and it is not sufficiently emphasized that the word "integral" now means something completely different than before. And now it turns out that the most important thing that is contained in the theory is acquired only furtively - through the back door, and the student encounters serious difficulties in his efforts to understand the essence of the matter. We prefer to call functions G(x) for which G0 (x) = f(x) not “indefinite integrals”, but antiderivatives of the function f(x). Then the main theorem can be formulated as follows:

The function F (x), which is an integral of the function f(x) with a constant lower and a variable upper limit x, is one of the antiderivatives of the function f(x).

We say "one of" the antiderivative functions for the reason that if G(x) is an antiderivative function of f(x), then it is immediately clear that any function of the form H(x) = G(x) + c (c - arbitrary constant) is also an antiderivative, since H0 (x) = G0 (x). The converse is also true. Two antiderivative functions G(x)

and H(x) can differ from one another only by a constant term. Indeed, the difference U(x) = G(x) − H(x) has U0 (x) = G0 (x) − H0 (x) = f(x) − f(x) = 0 as a derivative, i.e., That is, this difference is constant, since it is obvious that if the graph of a function is horizontal at each of its points, then the function itself, represented by the graph, must certainly be constant.

This leads to a very important rule for calculating the integral between a and b - assuming that we know some antiderivative function G(x) of the function f(x). According to our main

MAIN THEOREM OF ANALYSIS

theorem, function

there is also an antiderivative function of the function f(x). So F(x) =

G(x) + c, where c is a constant. The value of this constant will be determined,

if we take into account that F (a) = f(u) du = 0. This implies:

0 = G(a) + c, so c = −G(a). Then the definite integral between a and x identically satisfies the equality

F (x) = f(u) du = G(x) − G(a);

replacing x through b leads to the formula

f(u) du = G(b) − G(a),

regardless of which of the antiderivative functions was "launched". In other words: to calculate a certain in-

integral f(x) dx, it suffices to find a function G(x) for which

swarm G0 (x) = f(x), and then make the difference G(b) − G(a).

2. First applications. Integration of functions xr , cos x, sin x. arctg x function. Here it is impossible to give an exhaustive idea of ​​the role of the main theorem, and we confine ourselves to giving a few expressive examples. In problems encountered in mechanics and physics or in mathematics itself, it is very often necessary to calculate the numerical value of some definite integral. A direct attempt to find the integral as a limit can be insurmountably difficult. On the other hand, as we saw in § 3, any differentiation is carried out relatively easily, and it is possible to accumulate a very large number of differentiation formulas without difficulty. Each such formula G0 (x) = f(x), conversely, can be considered as a formula defining the antiderivative function G(x) of the function f(x).

Formula (3) allows using the known antiderivative function to calculate the integral of the function f(x) in some given interval.

If we, for example, want to find integrals of powers x2, x3, or xn in general, then the simplest thing is to proceed as indicated in § 1. By the power differentiation formula, the derivative of xn is nxn−1,

470 MATHEMATICAL ANALYSIS Ch. VIII

so the derivative of the function

G(x) = n x
	1 (n 6= -1)

there is a function

G0 (x) = n n + + 1 1 xn = xn .

xn+1

In this case, the function n + 1 is the antiderivative function

with respect to the function f(x) = xn , and therefore we immediately obtain the formula

x n dx = G(b) − G(a) = b n+1 − a n+1 . n + 1

This argument is incomparably simpler than the cumbersome procedure for directly calculating the integral as the limit of the sum.

As a more general case, we found in § 3 that for any rational s, both positive and negative, the derivative of the function xs is equal to sxs−1 , and therefore, for s = r + 1, the function

x r+1

has a derivative f(x) = G0 (x) = xr (we assume that r 6= −1,

x r+1

i.e. that s 6= 0). So the function r + 1 is the antiderivative function, or

"indefinite integral" of xr , and we get (for positive a and b and for r 6= −1) the formula

xr dx =	b r+1 − a r+1

In formula (4), one has to assume that the function xr under the integral is defined and continuous in the integration interval, so the point x = 0 must be excluded if r< 0. Вот потому мы и вынуждены допустить, что в этом случае a и b положительны.

If we set G(x) = − cos x, then we obtain G0 (x) = sin x, and hence the relation arises

sin xdx = -(cos a - cos 0) = 1 - cos a.

Similarly, if G(x) = sin x, then G0 (x) = cos x, and hence

cos xdx \u003d sin a - sin 0 \u003d sin a.

§ 5 MAIN THEOREM OF ANALYSIS 471

A particularly interesting result is obtained from the formula for differentiating the function arctg x:

	Since the function arctg x is antiderivative with respect to the function
							1+x2

then, based on formula (3), we can write

arctan b − arctan 0 = Z 0	1 + x2dx.

But arctan 0 = 0 (a zero value of the tangent corresponds to a zero value of the angle). So we have

arctg b = Z 0

1+x2

In particular,

meaning

tangent,

1, match

at 45◦, which in radian measure corresponds to

puts p . Thus, we

we get

wonderful

1 + x2dx.

shows

what area

schedule

1 + x 2 ranging from x = 0 to x =

1 is equal to a quarter of the area of ​​the unit

276. Area under the Cree

no circle.

within

3. Formula

Leibniz

1+x2

leads

for p . Latest Result

of the most beautiful

mathematical formulas discovered in the 17th century - to a sign-variable

to the Leibniz series, which allows calculating p:

4 p = 1 1 − 3 1 + 5 1 − 7 1 + 9 1 − 11 1 + . . .

+ symbol. . . should be understood in the sense that the sequence of finite "partial sums" obtained when the right-hand side of the

of equalities, only n terms of the sum are taken, tends to the limit p at

unlimited increase of n.

MATHEMATICAL ANALYSIS

To prove this remarkable formula, we need only recall the formula for the sum of a finite geometric progression

1 − qn = 1 + q + q2 + . . . + qn−1 ,

where the "residual term" Rn is expressed by the formula

Rn = (−1)n x 2n 2 .

Equality (8) can be integrated within the range from 0 to 1. Following rule a) from § 3, we must take the sum of the integrals of the individual terms on the right side. Based on (4) we know that

xm dx =

bm+1

− am+1

in particular, we get

xm dx =

from where, to

1+x2

1 − 3 +

And consequently,

− 7

+ . . . + (−1)n−1

2n − 1 + Tn ,

p R0

1+x2

Tn = (

According to formula (5), the left side of the form is

ly (9 ) is

difference between

and private sum

(−1)n−1

Sn = 1 -

− Sn = Tn . It remains to prove that Tn tends to zero as

increasing n. We have an inequality

x 2n 6 x2n .

1+x2

Recalling the formula (13) § 1, which establishes the inequality

f(x) dx 6 g(x) dx for f(x) 6 g(x) and a< b,

The concept of integration, and to some extent of differentiation, was well developed before the work of Newton and Leibniz. But it was absolutely necessary to make one very simple discovery in order to give impetus to the enormous evolution of the newly created mathematical analysis. Two apparently mutually non-contiguous limit processes, used one for differentiation, the other for integrating functions, turned out to be closely connected with each other. Indeed, they are mutually inverse operations, like such operations as addition and subtraction, multiplication and division. Differential and integral calculus are one thing.

The great achievement of Newton and Leibniz is that for the first time they clearly realized and used this basic theorem of analysis. Undoubtedly, their discovery lay in the direct path of natural scientific development, and it is not at all surprising that various persons came independently and almost simultaneously to a clear understanding of the above circumstance.

Rice. 274. Integral as a Function of the Upper Limit

In order to formulate the main theorem precisely, we consider the integral of a function ranging from a constant number a to a number x, which we will consider variable. In order not to confuse the upper limit of integration x with the variable appearing under the integral sign, we write the integral in the following form (see p. 459):

thus demonstrating our intention to study the integral as a function of its upper limit (Fig. 274). This function is the area under the curve from point to point. Sometimes an integral with a variable upper limit is called an "indefinite integral."

The main theorem of analysis reads as follows: The derivative of the indefinite integral (1) with respect to its upper limit x is equal to the value of the function at the point

In other words, the integration process leading from function to function is “destroyed” by the inverse process of differentiation applied to the function

Rice. 275. On the proof of the main theorem

On an intuitive basis, the proof of this proposition is not difficult. It is based on the interpretation of the integral as an area, and would be obscured if we tried to plot the function and interpret the derivative as the corresponding slope. Leaving aside the previously established geometric interpretation of the derivative, we will retain the geometric interpretation of the integral as an area, and we will become an analytical method to differentiate a function. Difference

there is simply the area under the curve between the limits (Fig. 275), and it is not difficult to understand that the numerical value of this area is enclosed between the numbers

where are (respectively, the largest and smallest values ​​of the function in the interval from to) Indeed, these products give the areas of two rectangles, one of which contains the curvilinear region under consideration, and the other is contained in it.

This implies:

Let us assume that the function is continuous, so that both quantities tend to the value of the function

at the point , i.e., to the value In this case, we can consider it proved that

The intuitive meaning of this result is that as the area increases, the rate of change of the area under the curve is equal to the height of the curve at x.

In some manuals, the content of this main theorem is obscured due to poorly chosen terminology. Namely, many authors first introduce the concept of a derivative, and then define the "indefinite integral" simply as the result of the operation inverse to differentiation: they say that a function is an indefinite integral of a function if

Thus, this way of presentation directly connects differentiation with the word "integral". It is only later that the concept of "definite integral" is introduced, treated as an area or as the limit of a sequence of sums, and it is not sufficiently emphasized that the word "integral" now means something completely different than before. And now it turns out that the most important thing that is contained in the theory is acquired only furtively - through the back door, and the student encounters serious difficulties in his efforts to understand the essence of the matter. We prefer functions for which we call not “indefinite integrals”, but antiderivative functions of a function. Then the main theorem can be formulated as follows:

A function that is an integral of a function with a constant lower and a variable upper limit x is one of the antiderivatives of the function

We say “one of” the antiderivative functions for the reason that if is an antiderivative function of then it is immediately clear that any function of the form (c is an arbitrary constant) is also an antiderivative, since the converse statement is also true. Two antiderivative functions can differ from one another only by a constant term. Indeed, the difference has as a derivative i.e. this difference is constant, since it is obvious that if the function graph in each

The concept of integration, and to some extent of differentiation, was well developed before the work of Newton and Leibniz. But it was absolutely necessary to make one very simple discovery in order to give impetus to the enormous evolution of the newly created mathematical analysis. Two apparently mutually non-contiguous limiting processes, used one for differentiation, the other for integrating functions, turned out to be closely connected with each other. Indeed, they are mutually inverse operations, like such operations as addition and subtraction, multiplication and division. Differential and integral calculus are one thing.

The great achievement of Newton and Leibniz is that for the first time they clearly recognized and used this the main theorem of analysis. Undoubtedly, their discovery lay in the direct path of natural scientific development, and it is not at all surprising that various persons came independently and almost simultaneously to a clear understanding of the above circumstance.

In order to formulate the main theorem exactly, we consider the integral of the function y=f(x) ranging from a constant number a to a number x, which we will consider variable. In order not to confuse the upper limit of integration x with the variable appearing under the integral sign, we write the integral in the following form (see p. 435):

thus demonstrating our intention to study the integral as a function of F(x) of its upper limit (Fig. 274). This function F(x) is the area under the curve y=f(u) from the point u = a to the point u=x. Sometimes the integral F(x) with a variable upper limit is called the "indefinite integral".

The main theorem of analysis reads as follows: The derivative of the indefinite integral (1) with respect to its upper limit x is equal to the value of the function f (u) at the point u = x:

F "(x) \u003d f (x).

In other words, the integration process leading from the function f(x) to the function F(x) is "destroyed" by the inverse process of differentiation applied to the function F(x).

On an intuitive basis, the proof of this proposition is not difficult. It is based on the interpretation of the integral F(x) as an area, and would be obscured if we tried to plot the function F(x) and interpret the derivative F"(x) as the corresponding slope. Leaving aside the previously established geometric interpretation of the derivative , we will keep the geometric interpretation of the integral F (x) as an area, and differentiate the function F (x) will become an analytical method.

F (x 1) - F (x)

is just the area under the curve y=f(u) between limits u = x 1 and u=x(Fig. 275), and it is easy to understand that the numerical value of this area is enclosed between the numbers (x 1 - x)m and (x 1 - x) M:

(x 1 - x)m≤F (x 1) - F (x) ≤(x 1 - x) M,

where M and m are respectively the largest and smallest values ​​of the function f (u) in the interval from u = x to u = x 1 . Indeed, these products give the areas of two rectangles, of which one contains the curvilinear region under consideration, and the other is contained in it.

This implies:

Suppose that the function f (u) is continuous, so that as x 1 tends to x, both quantities M and m tend to the value of the function f (u) at the point u \u003d x, i.e., to the value of f (x). In this case, it can be considered proven that

The intuitive meaning of this result is that as the rate of change of the area under the curve increases, y=f(x) equal to the height of the curve at x.

In some manuals, the content of this main theorem is obscured by poorly chosen terminology. Namely, many authors first introduce the concept of a derivative, and then define the "indefinite integral" simply as the result of the operation inverse to differentiation: they say that the function G (x) is an indefinite integral of the function f (x) if

G"(x) = f(x).

Thus, this way of presentation directly connects differentiation with the word "integral". It is only later that the concept of "definite integral" is introduced, treated as an area or as the limit of a sequence of sums, and it is not sufficiently emphasized that the word "integral" now means something completely different than before. And now it turns out that the most important thing that is contained in the theory is acquired only furtively from the back door, and the student encounters serious difficulties in his efforts to understand the essence of the matter. We prefer functions G(x) for which G "(x) \u003d f (x), call not "indefinite integrals", but antiderivative functions from the function f(x). Then the main theorem can be formulated as follows:

The function F (x), which is the integral of the function f (x) with a constant lower and variable upper limit x, is one of the antiderivatives of the function f (x).

We say "one of" the antiderivative functions for the reason that if G(x) is an antiderivative function of f(x), then it is immediately clear that any function of the form H(x) = G(x) + c(c is an arbitrary constant) is also an antiderivative, since H "(x) = G" (x). The converse is also true. The two antiderivative functions G(x) and H(x) can only differ from each other by a constant term. Indeed, the difference U(x) = G(x) - H(x) has as a derivative U "(x) \u003d G" (x) - H "(x) \u003d f (x) - f (x) \u003d 0, i.e. this difference is constant, since it is obvious that if the graph of a function is horizontal at each of its points, then the function itself, represented by the graph, must certainly be constant.