Uncertain integral its basic properties. The simplest properties of the integrals. Questions for self-test

In this article we will list the main properties certain integral. Most of these properties are proved based on the concepts of a certain integral of Riemann and Darbu.

The calculation of a specific integral is very often carried out using the first five properties, so we will refer to them if necessary. The remaining properties of a specific integral are mainly used to evaluate various expressions.

Before moving to the main properties of a specific integral, we agree that a does not exceed b.

For the function y \u003d f (x), defined at x \u003d a, equality is fair.

That is, the value of a specific integral with the coincidence limits of integration is zero. This property is a consequence of determining the integral of Riemann, since in this case each integral amount for any partitioning of the gap and any point selection is zero, since, therefore, the estimate of the integral sums is zero.

For the function integrated on the segment is performed .

In other words, when changing the upper and lower limits of integration in places, the value of a specific integral changes to the opposite. This property of a specific integral also follows from the concept of the Riemann integral, only the numbering of the splitting of the segment should be started from the point x \u003d b.

For integrated on the segment of the functions y \u003d f (x) and y \u003d g (x).

Evidence.

We write the integrated amount of the function For this splitting of the segment and this selection of points:

where and is the integral sums of functions y \u003d f (x) and y \u003d g (x) for this splitting of the segment, respectively.

Moving to the limit when We obtain that, by definition of the Riemann integral, it is equivalent to the approval of the proven properties.

A permanent multiplier can be taken out of a certain integral sign. That is, for the function integrated on the segment Y \u003d F (X) and an arbitrary number K, equality is true .

The proof of this property of a specific integral is absolutely similar to the previous one:


Let the function y \u003d f (x) integrable on the interval x, and and then .

This property is fair both for and for or.

Proof can be carried out by relying on previous properties of a specific integral.

If the function is integrated on the segment, then it is integrated and on any inner segment.

The proof is based on the property of Darboux: if you add new points to the existing splitting of the segment, then the lower amount of Darboux will not decrease, and the upper one will not increase.

If the function y \u003d f (x) is integrated on the segment and for any value of the argument, then .

This property is proved through the determination of the Riemann integral: any integral amount for any selection of the separation points of the segment and points will be non-negative (not positive).

Corollary.

For integrated on the segment of functions y \u003d f (x) and y \u003d g (x), inequalities are valid:


This statement means that it is permissible to integrate inequalities. We will use this consequence in the proof of the following properties.

Let the function y \u003d f (x) integrate on the segment, then the inequality is true .

Evidence.

It's obvious that . In the previous property, we found out that inequality can be integrated so far, so . This dual inequality can be written as .

Let the functions y \u003d f (x) and y \u003d g (x) integrable on the segment and for any value of the argument, then where and .

Proof is carried out similarly. As m and m - the smallest and the greatest value functions y \u003d f (x) on the segment, then . Doming the double inequality on the nonnegative function y \u003d g (x) leads us to the next double inequality. Integrating it on the segment, we will come to the proven statement.

Primed function and indefinite integral

Fact 1. Integration - action, inverse differentiation, namely, restoring the function according to a known derivative of this function. Function restored F.(x.) Called predo-shaped For function f.(x.).

Definition 1. Function F.(x. f.(x.) at some interval X.if for all values x. equality is performed from this gap F. "(x.)=f.(x.), that is, this feature f.(x.) is derived from pred-like function F.(x.). .

For example, a function F.(x.) \u003d SIN. x. is a primary for function f.(x.) \u003d COS. x. on the whole numerical straight, since with any value of the IKSA (sin. x.) "\u003d (COS x.) .

Definition 2. Uncertainly integral function f.(x.) It is called the totality of all its primitive. This uses recording

∫

f.(x.)dX.

,

where sign ∫ called the integral sign, function f.(x.) - a replacement function, and f.(x.)dX. - A concrete expression.

Thus, if F.(x.) - some kind of primary for f.(x.), T.

∫

f.(x.)dX. = F.(x.) +C.

where C. - arbitrary constant (constant).

To understand the meaning of many primitive functions as an indefinite integral, the following analogy is appropriate. Let there be a door (traditional wooden door). Its function is "to be a door." And what is the door made from? From wood. Therefore, a multitude of primitive integrated function "Be the door", that is, it is an indefinite integral, is the function "Being + C", where C is a constant, which in this context may indicate, for example, a tree of wood. Just as the door is made of wood using some tools, the derivative of the "made" function from the primitive function with the formulas that we learned by studying the derivative .

Then the table of the functions of common objects and the corresponding primitive ("to be the door" - "be tree", "be a spoon" - "be metal", etc.) is similar to the table of the main indefinite integrals, which will be shown slightly below. The table of uncertain integrals lists common functions with the indication of the primordial, of which these functions are made. In terms of the tasks to find a indefinite integral, such integrants are given, which without particular gravity can be integrated directly, that is, on the table of uncertain integrals. In the tasks, it is necessary to pre-convert to the tasks to preform so that you can use table integrals.

Fact 2. Restoring the function as a primitive, we must take into account an arbitrary constant (constant) C., so as not to write a list of primitive with different constants from 1 to infinity, you need to record many of the primitive with an arbitrary constant C.For example, as follows: 5 x.³ + s. So, an arbitrary constant (constant) enters the expression of primitive, since the primitive can be a function, for example, 5 x.³ + 4 or 5 x.³ + 3 and with differentiation 4 or 3, or any other constant is applied to zero.

We will put the integration task: for this function f.(x.) find such a function F.(x.), derivative of which equal f.(x.).

Example 1.Find a variety of features

Decision. For this feature, the function is function

Function F.(x.) called primitive for function f.(x.) if derivative F.(x.) Equal f.(x.), or that the same, differential F.(x.) Raven f.(x.) dX..

(2)

Consequently, the function is primitive for a function. However, it is not the only primary for. They also serve as functions

where FROM - Arbitrary constant. This can be seen differentiation.

Thus, if there is one first primary for the function, then it has an infinite multitude of primitive, differing in permanent term. All the primary functions are written in the above form. This follows from the following theorem.

Theorem (formal statement of fact 2).If a F.(x.) - Valid for function f.(x.) at some interval H., then any other primitive for f.(x.) At the same gap can be presented in the form F.(x.) + C.where FROM- Arbitrary constant.

In the following example, we already appeal to the integral table, which will be given in paragraph 3, after the properties of an indefinable integral. We do it before familiarization with the entire table, so that the essence of the foregoing is understood. And after the table and properties we will use them when integrating in all fullness.

Example 2.Find multiple features:

Decision. We find the sets of primitive functions, of which "these functions are made". When mentioning the formulas from the integral table, simply accept that there are such formulas, and we will study the table of uncertain integrals to be completely further.

1) applying formula (7) from the integral table with n. \u003d 3, we get

2) using formula (10) from the integral table with n. \u003d 1/3, we have

3) as

then by formula (7) when n. \u003d -1/4 Find

Under the sign of the integral write not the function itself f. , and her work on differential dX. . This is done primarily in order to indicate which variable is looking for a primitive. For example,

, ;

here, in both cases, the integrand function is equal, but its indefinite integrals in the considered cases are different. In the first case, this feature is considered as a function from a variable x. , and in the second - as a function from z. .

The process of finding an indefinable integral function is called integrating this function.

Geometric meaning of an indefinite integral

Let it be required to find a curve y \u003d f (x) And we already know that the tangent of the tilt angle at each of its point is the specified function f (x) The abscissions of this point.

According to the geometric meaning of the derivative, tangent tilt angle at this point of the curve y \u003d f (x) equal to the value of the derivative F "(x). So you need to find such a function F (x), for which F "(x) \u003d f (x). Function required in the task F (x) is a primary one f (x). The condition of the problem satisfies not one curve, but the family of curves. y \u003d f (x) - one of such curves, and every other curve can be obtained from her parallel transfer along the axis Oy..

Let's call a graph of a primitive function from f (x) integral curve. If a F "(x) \u003d f (x)then the graph of the function y \u003d f (x) There is an integral curve.

Fact 3. An uncertain integral is geometrically represented by the seven of all integrated curves As in the figure below. The remoteness of each curve from the start of coordinates is determined by an arbitrary constant (constant) integration C..

Properties of an indefinite integral

Fact 4. Theorem 1. The derivative of an indefinite integral is equal to the integrand function, and its differential is a source expression.

Fact 5. Theorem 2. Unexposed integral from differential function f.(x.) Equal function f.(x.) with an accuracy of a permanent term .

(3)

Theorems 1 and 2 show that differentiation and integration are mutually reverse operations.

Fact 6. Theorem 3. A constant multiplier in the integrand can be made for a sign of an indefinite integral .

Let the function y. = f.(x. ) Defined on the segment [ a., b. ], a. < b. . Perform the following operations:

1) Drain [ a., b. ] Points a. = x. 0 < x. 1 < ... < x. i.- 1 < x. i. < ... < x. n. = b. on the n. Partial segments [ x. 0 , x. 1 ], [x. 1 , x. 2 ], ..., [x. i.- 1 , x. i. ], ..., [x. n.- 1 , x. n. ];

2) in each of the partial segments [ x. i.- 1 , x. i. ], i. = 1, 2, ... n., Select an arbitrary point and calculate the value of the function at this point: f.(z I. ) ;

3) find works f.(z I. ) · Δ x. i. where - the length of the partial segment [ x. i.- 1 , x. i. ], i. = 1, 2, ... n.;

4) make up integral sumfunctions y. = f.(x. ) on the segment [ a., b. ]:

FROM geometrical point Vision This sum σ is the sum of the areas of rectangles, the grounds of which are partial segments [ x. 0 , x. 1 ], [x. 1 , x. 2 ], ..., [x. i.- 1 , x. i. ], ..., [x. n.- 1 , x. n. ] and heights are equal f.(z. 1 ) , f.(z. 2 ), ..., f.(z N. ) Accordingly (Fig. 1). Denote by λ The length of the greatest partial segment:

5) We find the limit of the integrated amount when λ → 0.

Definition. If there is a finite integrated amount (1) and it does not depend on the method of splitting the segment [ a., b. ] on partial segments, nor from selecting points z I. in them, then this limit is called certain integral from function y. = f.(x. ) on the segment [ a., b. ] And denotes

In this way,

In this case, the function f.(x. ) Called integrable on the [ a., b. ]. Numbers a. and b. referred to as the bottom and top of the integration limits, f.(x. ) - the integrand function, f.(x. ) dX. - a concintive expression, x. - integration variable; section [ a., b. ] It is called integration interval.

Theorem 1.If the function y. = f.(x. ) continuous on the segment [ a., b. ], then it is integrated on this segment.

A certain integral with the same integration limits is zero:

If a a. > b. , then, by definition, we believe

2. Geometric meaning of a specific integral

Let on the segment [ a., b. ] Continuous non-negative function y. = f.(x. ) . Curvilinear trapeziumcalled a figure limited from the top of the function schedule y. = f.(x. ), below - the axis oh, left and right - direct x \u003d A. and x \u003d B. (Fig. 2).

Certain integral from non-negative function y. = f.(x. ) from a geometric point of view is equal to the area of \u200b\u200bcurvilinear trapezium, limited on top of the graph y. = f.(x. ), left and right - straight cuts x \u003d A. and x \u003d B. , below - the segment of the axis oh.

3. The main properties of a specific integral

1. The value of a specific integral does not depend on the designation of the integration variable:

2. A permanent multiplier can be taken out of a certain integral sign:

3. A specific integral from the algebraic amount of two functions is equal to the algebraic sum of certain integrals from these functions:

4.If function y. = f.(x. ) integrate on [ a., b. ] I. a. < b. < c. T.

5. (average theorem). If the function y. = f.(x. ) continuous on the segment [ a., b. ], then on this segment there is a point, such that

4. Newton Labitsa Formula

Theorem 2.If the function y. = f.(x. ) continuous on the segment [ a., b. ] I. F.(x.) - some kind of pre-shaped on this segment, then the following formula is true:

which is called newton's formula laboratory. Difference F.(b.) - F.(a.) It is customary to record as follows:

where the character is called a dual substitution sign.

Thus, formula (2) can be written in the form:

Example 1. Calculate integral

Decision. For the integrated function f.(x. ) = x. 2 arbitrary primitive has the form

Since in the formula Newton labnica, you can use any primitive, then to calculate the integral, take a primitive, having the simplest view:

5. Replacing the variable in a specific integral

Theorem 3. Let the function y. = f.(x. ) continuous on the segment [ a., b. ]. If a:

1) Function x. = φ ( t.) and its derivative φ "( t.) continuous with;

2) multiple function values x. = φ ( t.) when it is a segment [ a., b. ];

3) φ ( a.) = a., φ ( b.) = b.then the formula is valid

which is called the formula for replacing the variable in a specific integral .

Unlike an uncertain integral, in this case not necessary Return to the initial integration variable - it is enough to find new limits of integration α and β (for this it is necessary to solve relatively variable t. equations φ ( t.) = a. and φ ( t.) = b.).

Instead of substitution x. = φ ( t.) You can use substitution t. = g.(x. ). In this case, finding new integration limits by variable t.simplified: α \u003d g.(a.) , β = g.(b.) .

Example 2.. Calculate integral

Decision. We introduce a new variable by the formula. Erecting both parts of equality in the square, we get 1 + x \u003d. t. 2 From! x \u003d. t. 2 - 1, dX. = (t. 2 - 1)"dt.= 2tDT. . We find new integration limits. To do this, we will substitute the old limits in the formula x \u003d. 3 I. x \u003d. 8. We get: where t.\u003d 2 and α \u003d 2; From! t.= 3 and β \u003d 3. So

Example 3. Calculate

Decision. Let be u. \u003d LN. x. then v. = x. . By formula (4)

The basic integration formulas are obtained by appealing the formula for derivatives, so before the study of the topic under consideration, the differentiation formulas should be repeated 1 of the main functions (i.e. remember the derivatives table).

Getting acquainted with the concept of primitive, the definition of an indefinable integral and comparing differentiation and integration operations, students should pay attention to the fact that the operation of integration is meaningful, because Gives an infinite many primitive on the segment under consideration. However, the task of finding only one primitive is solved, because All primary this feature differ from each other for a permanent amount.

where C. - Arbitrary value 2.

Questions for self-test.

Give the definition of a primitive function.

What is called an uncertain integral?

What is the integrand?

What is a concintive expression?

Specify the geometrical meaning of the family of primitive functions.

6. Find the curve passing through the point

2. Properties of an indefinite integral.

Table of simplest integrals

Here, students should explore the following properties of an indefinite integral.

Property 1. The derivative of an indefinite integral is equal to the integrated 3 function (by definition)

Property 2. Differential from the integral is equal to the image

those. If the differential sign is facing the integral sign, they are mutually destroyed.

Property 3. If the integral sign is facing the differential sign, they are mutually destroyed, and an arbitrary constant value is added to the function.

Property4. The difference of two primitive one and the same function is the value constant.

Property 5. A permanent multiplier can be taken out of the integral sign

where BUT - constant number.

By the way, this property is easily proved by the differentiation of both parts of equality (2.4) taking into account the properties 2.

Property 6. The integral from the amount (difference) of the function is equal to the amount (difference) of the integrals from these functions (if they exist)

This property is also easily proved by differentiation.

Natural generalization Properties 6

. (2.6)

Considering the integration as an action, reverse differential, directly from the table of simple derivatives, you can get a table with the following simple integrals.

Table of the simplest indefinite integrals

1., where, (2.7)

2., where, (2.8)

4., where ,, (2.10)

9. , (2.15)

10. . (2.16)

Formulas (2.7) - (2.16) of the simplest indefinite integrals should be learned by heart. Their knowledge is necessary, but not enough to learn how to integrate. Sustainable skills in integration are achieved only by solving a sufficiently large number of tasks (usually about 150-100 examples of various types).

Below are examples of simplifying integrals by converting them to the sum of known integrals (2.7) - (2.16) from the above table.

Example 1.

.

This article tells in detail the basic properties of a particular integral. They are proved by the concept of the integral of Riemann and Darbu. Calculation of a specific integral passes, thanks to 5 properties. The remaining of them are used to estimate various expressions.

Before moving to the main properties of a specific integral, it is necessary to make sure that A does not exceed b.

The main properties of a specific integral

Definition 1.

The function y \u003d f (x), defined at x \u003d A, similar to the equitable equality ∫ a a f (x) d x \u003d 0.

Proof 1.

From here we see that the integral value with the coinciding limits is zero. This is a consequence of the Riemann integral, because each integral sum Σ for any partition at the interval [A; A] and any selection of points ζ i is zero, because x i - x i - 1 \u003d 0, i \u003d 1, 2 ,. . . , n, it means that we obtain that the limit of integral functions is zero.

Definition 2.

For a function integrated on the segment [a; b], the condition ∫ a b f (x) d x \u003d - ∫ b a f (x) d x is satisfied.

Proof 2.

In other words, if you change the upper and lower limit of integration in places, the integral value will change the value to the opposite. This property is taken from the integral of Riemann. However, the numbering of the splitting of the segment comes from the point x \u003d b.

Definition 3.

∫ a b f x ± g (x) d x \u003d ∫ a b f (x) d x ± ∫ a b g (x) d x is used for integrable functions like y \u003d f (x) and y \u003d g (x) defined on the segment [a; b].

Proof 3.

Record the integrated sum of the function y \u003d f (x) ± g (x) to split into segments with a given choice of points ζ i: σ \u003d Σ i \u003d 1 nf ζ i ± g ζ i · xi - xi - 1 \u003d \u003d σ i \u003d 1 nf (ζ i) · xi - xi - 1 ± σ i \u003d 1 ng ζ i · xi - xi - 1 \u003d σ f ± σ g

where σ f and σ g are the integral sums of the functions y \u003d f (x) and y \u003d g (x) to split the segment. After transition to the limit at λ \u003d m a x i \u003d 1, 2 ,. . . , N (x i - x i - 1) → 0 We obtain that Lim Λ → 0 Σ \u003d Lim Λ → 0 Σ F ± Σ G \u003d Lim Λ → 0 Σ g ± Lim Λ → 0 Σ g.

From the definition of Riemann, this expression is equivalent.

Definition 4.

Reaching a constant factor for a sign of a certain integral. Integrable function from the interval [a; b] with an arbitrary value K has a fair inequality of the form ∫ a b k · f (x) d x \u003d k · ∫ a b f (x) d x.

Proof 4.

Proof of the properties of a specific integral similarly to the previous one:

σ \u003d σ i \u003d 1 nk · f ζ i · (xi - xi - 1) \u003d k · σ i \u003d 1 nf ζ i · (xi - xi - 1) \u003d k · σ f ⇒ lim λ → 0 σ \u003d Lim Λ → 0 (k · Σ f) \u003d k · lim λ → 0 Σ F ⇒ ∫ ABK · F (X) DX \u003d k · ∫ ABF (X) DX

Definition 5.

If the function of the form y \u003d f (x) is integrated on the interval x with a ∈ x, b ∈ X, we obtain that ∫ a b f (x) d x \u003d ∫ a c f (x) d x + ∫ c b f (x) d x.

Proof 5.

The property is considered valid for c ∈ A; b, for c ≤ a and c ≥ b. Proof is carried out similarly to previous properties.

Definition 6.

When the function has the ability to be integrated from the segment [A; b], then it is done for any inner segment C; d ∈ A; b.

Proof 6.

The proof is based on the Darbé property: if the existing splitting of the segment is added to add points, then the lower amount of Darboux will not decrease, and the upper will not increase.

Definition 7.

When the function is integrated to [A; b] from f (x) ≥ 0 f (x) ≤ 0 at any meaning x ∈ A; b, then we obtain that ∫ a b f (x) d x ≥ 0 ∫ A b f (x) ≤ 0.

The property can be proven using the determination of the Riemann integral: any integral amount for any selection of the separation points of the segment and points ζ i with the condition that f (x) ≥ 0 f (x) ≤ 0, we obtain nonnegative.

Proof 7.

If the functions y \u003d f (x) and y \u003d g (x) are integrable on the segment [a; b], then the following inequalities are considered fair:

∫ a b f (x) d x ≤ ∫ a b g (x) d x, e s l and f (x) ≤ g (x) ∀ x ∈ A; b ∫ a b f (x) d x ≥ ∫ a b g (x) d x, e s l and f (x) ≥ g (x) ∀ x ∈ A; B.

Thanks to the statement, we know that integration is permissible. This investigation will be used in the proof of other properties.

Definition 8.

With the integrable function y \u003d f (x) from the segment [a; b] We have a fair inequality of the form ∫ A B f (x) d x ≤ ∫ a b f (x) d x.

Proof 8.

We have that - f (x) ≤ f (x) ≤ f (x). From the previous property, it was obtained that inequality can be integrated and corresponds to it inequality of the form - ∫ a b f (x) d x ≤ ∫ a b f (x) d x ≤ ∫ a b f (x) d x. This double inequality can be recorded in the other form: ∫ a b f (x) d x ≤ ∫ a b f (x) d x.

Definition 9.

When the functions y \u003d f (x) and y \u003d g (x) are integrated from the segment [a; b] for g (x) ≥ 0 at any x ∈ A; b, we obtain the inequality of the form m · ∫ a b g (x) d x ≤ ∫ a b f (x) · g (x) d x ≤ m · ∫ a b g (x) d x, where m \u003d m i n x ∈ A; b f (x) and m \u003d m a x x ∈ A; B f (x).

Proof 9.

Similarly proof is proof. M and M are considered the greatest and smallest value of the function y \u003d f (x), determined from the segment [A; b], then m ≤ f (x) ≤ m. It is necessary to multiply the double inequality on the function y \u003d g (x), which will give the value of the double inequality of the form M · G (x) ≤ f (x) · g (x) ≤ m · g (x). It is necessary to integrate it on the segment [a; b], then we will get a proven statement.

Consequence: At G (x) \u003d 1, the inequality takes the form M · b - a ≤ ∫ a b f (x) d x ≤ m · (b - a).

The first middle formula

Definition 10.

At y \u003d f (x) integrated on the segment [a; b] with m \u003d m i n x ∈ A; b f (x) and m \u003d m a x x ∈ A; b f (x) there is a number μ ∈ M; M, which is suitable for ∫ a b f (x) d x \u003d μ · b - a.

Consequence: When the function y \u003d f (x) is continuous from the segment [a; b], then there is such a number C ∈ A; b, which satisfies the equality ∫ a b f (x) d x \u003d f (c) · b - a.

The first middle formula in generalized formula

Definition 11.

When the functions y \u003d f (x) and y \u003d g (x) are integrable from the segment [A; b] with m \u003d m i n x ∈ A; b f (x) and m \u003d m a x x ∈ A; b f (x), and g (x)\u003e 0 for any meaning x ∈ A; b. From here we have that there is a number μ ∈ M; M, which satisfies the equality ∫ a b f (x) · g (x) d x \u003d μ · ∫ a b g (x) d x.

Second medium formula

Definition 12.

When the function y \u003d f (x) is an integrable of a segment [a; b], and y \u003d g (x) is monotonous, then there is a number that C ∈ A; b, where we obtain a fair equality of the form ∫ a b f (x) · g (x) d x \u003d g (a) · ∫ a C f (x) d x + g (b) · ∫ c b f (x) d x

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