Integrals for teapots: how to break the rules of calculation, explanation. Just the power of the integrals

The data of power are victorious for the creation of the transformation of the integral with the method of yogo reduction to one of the elementary integrals and the farthest calculation.

1. Pokhіdna neznachennoy іntegrа dоrіvnyuє pіdіntegrаlіnіy funkії:

2. The differential of the undefined integral is better than the pintegral virase:

3. Non-values ​​of the integral in the form of the differential of the current function of the healthy sum of the functions and the sufficient post:

4. A constant multiplier can be blamed for the sign of the integral:

Moreover, a ≠ 0

5. Integral of the sum (retail) of the additional sum (retail) of the integral:

6. Powerfulness - combination of powers 4 and 5:

Moreover, a ≠ 0 ˄ b ≠ 0

7. The power of the invariance of the undefined integral:

Yakscho something

8. Power:

Yakscho something

In fact, power is given to us, we call it integration for the additional method of replacing the change, which is more reportable in the attacking division.

Let's look at the example:

On the back of the head, we scored power 5, then power 4, scored the first table and took the result.

The algorithm of our online calculator іntegrіlіv pіdtrimuє sі vіlіchenі vyshe vіsshego vіstіvі and і easily znayde rіshenі prіshennia іn ііntegrаіv.

The main tasks of the differential calculationє familiarity f'(x) or differential df=f'(x)dx functions f(x). The problem is wrapped in the integral number. For a given function f(x) it is necessary to know such a function F(x), what F'(x)=f(x) or dF(x)=F'(x)dx=f(x)dx.

in such a manner, the main tasks of the integral calculationє function upgrade F(x) for the venerable (differential) function. Integral calculation of the largest numerical additions to geometry, mechanics, physics and technology. Vono gives a wild method of knowing the area, obsyagіv, centers of importance just.

Appointment. FunctionF(x), , is called primary for the functionf(x) by a multiple of X, as it is differentiated for whatever iF'(x)=f(x) ordF(x)=f(x)dx.

Theorem. Be-yaka bezperervna on vіdrіzku [a;b] functionf(x) May I choose the first oneF(x).

Theorem. YakschoF 1 (x) thatF 2 (x) – two different primary one and the same functionsf(x) on impersonal x, then they stink one by one with a constant dodank, that is.F 2 (x)=F1x)+C de C - postyna.

Insignificance of the integral, yoga of dominance.

Appointment. SukupnistnostF(x)+Three primary functionsf(x) on the multiplier X is called an insignificant integral and is assigned:

- (1)

Formula (1) f(x)dx called pidintegral virase,f(x) - integrating function, x - variable integration, a H - post-ynoї іntegratsії.

Let's look at the authorities not sing integral, yakі z yoga appointment.

1. Departure from the undefined integral to the pre-integral function, the differential of the undefined integral to the pre-integral virase:

that .

2. The non-values ​​of the integral in the differential of the current function are good for the sum of the functions and the sufficient post:

3. The constant multiplier a (a≠0) can be blamed for the sign of the undefined integral:

4. The non-values ​​of the integral in the sum of the algebra of the end number of functions are the additional sum of the algebra of the integrals in the form of these functions:

5. YakschoF(x) – primary functionf(x), then:

6 (invariance of integration formulas). Whether the formula of integration saves its own sight, if it changes the integration, replace it with some function of the variable, which differentiates:

deu - function to be differentiated.

Table of non-significant integrals.

Navigable basic rules for integrating functions.

Navigable table of basic non-significant integrals(It is significant that here, like i in the differential calculation, the letter u can you name an independent change (u=x), so the function as an independent change (u=u(x)).)

(n≠-1). (a>0, a≠1). (a≠0). (a≠0). (|u|> |a|).(|u|< |a|).

Integrals 1 - 17 name tabular.

Deyakі z inducing more formulas of the tables of integrals, yakі mayut analogue of the tables of similar ones, they are overturned by the derivations of their right parts.

Replacement of change and integration by parts in an undefined integral.

Integration of substations (replacement of replacements). Let it be necessary to calculate the integral

, which is not tabular. The essence of the substitution method lies in the fact that in the integral change X replace the change t behind the formula x = φ (t), stars dx=φ'(t)dt.

Theorem. Come on functionx = φ (t) that differentiation is assigned to the real multiplier T and let X is the impersonal value of the function, to which the function is assignedf(x). Same as on the X multiplier functionf(

The solution of integrations is easier, but more difficult. This article is for those who want to learn to understand the integration, but do not know anything about them or anything. Integral... Need more wine? How to calculate yoga? What is the difference between the integrals?

If only you can see the integration of the integral - put it in the shape of the icon of the integral to the right of the most important places, then we kindly ask you! Find out how to solve the simplest and the other integrals and why you can’t do without them in mathematics.

We understand « integral »

Integration was in the home Ancient Egypt. Obviously, not in a modern way, but all the same. In the meantime, mathematicians have written a lot of books on this topic. Particularly noted newton і Leibnitz but the essence of the speeches did not change.

How to understand integrated from zero? Niyak! For the sake of understanding, all the same, one needs basic knowledge of the basics mathematical analysis. Vіdomosti about interi and pokhіdnі, necessary and for the understanding of integrations, yet we have blozі.

Non-significance integral

Let us have it as a function f(x) .

The non-significant integral of the function f(x) such a function is called F(x) , similar to the old functions f(x) .

Tobto іntegral - tse pokhіdna navpak or primary. Before speech, about those, how to count the bad ones, read in our article.

Primary source for everyone uninterrupted functions. Also, the symbol of a constant is often added to the primary one, the shards of similar functions, which differ by a constant, are avoided. The process of learning the integral is called integration.

Simple example:

To avoid violating the primary elementary functions, manually add them to the table and use the already prepared values.

Povna table of integrals for students

Integral value

To the right of the understanding of the integral, we may be to the right of inexorably small quantities. Integral additionally calculate the area of ​​the figure, the mass of a heterogeneous body, passages in uneven Russia, and a rich other. A trace of memory, that the integral is the sum of an infinitely large number of infinitely small additions.

As a butt, you can see the schedule as a function.

How to know the area of ​​a figure surrounded by a graph of a function? For the help of the integral! Rozіb'єmo curvilinear trapezium, surrounded by axes of coordinates and a graph of the function, on infinitely small ribs. In this way, the figure appears to be divided into thin pillars. The sum of the squares of the stovpchikiv and we become the square of the trapezium. But remember that such an calculation will give a clear result. However, if there are less and already there will be charges, then there will be more exact charges. If we change their flooring, if the dovzhina will sag to zero, then the sum of the area of ​​the vіdrіzkіv will sag to the area of ​​the figure. Tse i є pevny integrand, which is written like this:

The points a and b are called integration boundaries.

« Integral »

Before speech! For our readers at once a discount of 10% on some kind of robot

Rules for calculating integrals for teapots

Power of the undefined integral

How to solve the insignificances of the integral? Here we can look at the power of the unmeasured integral, like to become in good fortune with perfect applications.

• Pokhіdna vіd іntegrа dоrіvnyuє pіdіntegrіalіy funktsії:

• The constant can be blamed on the integral sign:

• Іntegrаl vіd sumi dorivnyuє sumі іntegrіlіv. The same is true for retail:

The power of the singing integral

• Linearity:

• The sign of the integral is changed to remember the points between integration:

• At whatever points a, bі h:

We have already explained that the song integral is the sum between the sums. Ale, how to take into account the specific meaning of the pіd hour of the virіshennya butt? For which one use the Newton-Leibniz formula:

Apply the perfection of integration

Below we look at the inconsistencies of the integral and apply it to the solutions. It is proponable to independently expand in the subtleties of the decision, but if it is unreasonable, put the question in the comments.

To consolidate the material, take a look at the video about those, how integrals are violated in practice. Do not fall into the void, as the integral is not given to the eye. Return to a professional service for students curvilinear integral along the closed surface you will be given strength.

At the statistics, we override the main power of the singing integral. The greater part of these powers is brought on the basis of understanding the singing integral of Rimmann and Darboux.

Calculation of the first integral is often carried out among the first five powers, so we will ask for the demand. The rest of the power of the singing integral is mainly used to evaluate different viruses.

First down go to the main powers of the sing integral, we think that a does not change b.

For the function y = f(x) , assigned for x = a, equality is valid.

That is why the values ​​of the sing integral z zbіgayutsya interami іntegruvannya doіvnyuє zero. This power is consequently assigned to the Riemann integral, to that in which case the skin integral sum for any kind of splitting the gap and whether or not the choice of the point is equal to zero, to that, then, the boundary of the integral sums is zero.

For a function that can be integrated on a widget, win .

In other words, when changing the upper and lower between the integration points, the value of the sing integral changes on the length. The power of the sing integral is also clear from the understanding of the Riemann integral, it is less the numbering of the split and the next start from the point x = b.

for integrating the functions y = f(x) and y = g(x) on a cross-sectional basis.

Bringing.

We write down the integral sum of the function for a given connection, a window and a choice of points:

de-integral sum of functions y = f(x) і y = g(x) for a given distribution in the opposite case.

Passing to the border with we take into account that the designation of the Riemann integral is equal to the assertion of the brought value.

The constant multiplier can be blamed for the sign of the sing integral. Therefore, for a function y = f(x) integrated on a cross-section and an appropriate number k, equality is true .

The proof of the sovereignty of the sing integral is absolutely similar to the previous one:


Let the function y = f(x) be integrated over the interval X, moreover and then .

Tsya vlastіvіst fair as for, so for or.

The proof can be carried out by spiraling on the forward power of the sing integral.

Just as the function is integrated on the windshield, it is also integrated to the internal windshield.

The proof rests on the power of Darboux's sum: if you add new points to the obvious breakdown, then the lower sum of Darboux will not change, and the upper one will not increase.

If the function y = f(x) is integrated on the subdivision y of any value of the argument , then .

Tsya power is brought through the designation of Riemann's integral: whether the sum is integral for whether there is a choice between a point of division and a point when it is invisible (not positive).

Consequence.

For integrating the functions y = f(x) and y = g(x) on the subsection, the following inconsistencies are true:


Tse firm means that the integration of irregularities is permissible. In the meantime, we are worthy of the hour to prove the impending dominance.

Let the function y = f(x) be integrated on the reverse, then the non-uniformity is valid .

Bringing.

Obviously what . It was stated in front of the authorities that inconsistency can be integrated term by term, it is fair . You can write down how .

Let the functions y = f(x) and y = g(x) be integrated on the subsection i for any value of the argument , then , de that .

The proof is carried out in a similar way. So m and M are the least the most significant function y = f(x) on the opposite side, then . Multiplying the subvariant unevenness by the unknown function y = g(x) is brought to the offensive subvariant unevenness. Integrating yoga into the air, we will come to a firmness, what to bring.

Consequence.

If we take g(x) = 1, then I will see the unevenness in the future .

First formula for the average.

Let the function y = f(x) be integrated on the reverse, and then it is the same number that .

Consequence.

If the function y = f(x) is interruptless, then there is a number such that .

The first formula of the mean value of the narrowed form.

Let the functions y = f(x) and y = g(x) be integrated on the reverse side, і , and g(x) > 0 no matter what the value of the argument . Then you have such a number that .

Another formula for the average.

In addition, the function y = f(x) is integrated, and y = g(x) is monotonic, so there is a number such that equality is valid .

The data of power are victorious for the creation of the transformation of the integral with the method of yogo reduction to one of the elementary integrals and the farthest calculation.

1. Pokhіdna neznachennoy іntegrа dоrіvnyuє pіdіntegrаlіnіy funkії:

2. The differential of the undefined integral is better than the pintegral virase:

3. Non-values ​​of the integral in the form of the differential of the current function of the healthy sum of the functions and the sufficient post:

4. A constant multiplier can be blamed for the sign of the integral:

Moreover, a ≠ 0

5. Integral of the sum (retail) of the additional sum (retail) of the integral:

6. Powerfulness - combination of powers 4 and 5:

Moreover, a ≠ 0 ˄ b ≠ 0

7. The power of the invariance of the undefined integral:

Yakscho something

8. Power:

Yakscho something

In fact, power is given to us, we call it integration for the additional method of replacing the change, which is more reportable in the attacking division.

Let's look at the example:

On the back of the head, we scored power 5, then power 4, scored the first table and took the result.

The algorithm of our online calculator іntegrіlіv pіdtrimuє sі vіlіchenі vyshe vіsshego vіstіvі and і easily znayde rіshenі prіshennia іn ііntegrаіv.