Method for replacing a replacement part with an unassigned integral butt. Integration with the replacement path (substitution method). Method for replacing a variable in an unidentified integral

Replacing the variable for an unidentified integral. Formula for the transformation of differentials. Application of integration. Apply linear substitutions.

Zmist

Div. also: Table of non-significant integrals
Basic elementary functions and their powers

Replacement method

By additionally replacing the variable, you can calculate simple integrals and, in some cases, simplify the calculation of fold ones.

The method of replacing the exchangeable one with the output variable integration, Let it be x, let’s move on to another change, which is significant like t. In this case, we respect that the changes x and t are related to the actual relationship x = x (t), or t = t (x). For example, x = ln t, x = sint, t = 2 x + 1, etc. Our task is to select such a density between x and t, so that the output integral either returns to the tabular one, or becomes more simple.

Basic formula for replacing meat

Let's look at what stands under the integral sign. It consists of the creation of an integral function, which is significant as f (x) and differential dx: . Let us move on to a new change t by choosing the relationship x = x (t). Then we can determine the function f (x) and differential dx through variable t.

To express the integral function f (x) through the change t you just need to substitute the change x in the selected relationship x = x (t).

The reversal of the differential is calculated as follows:
.
Then the differential dx is the same as the differential x in t to the differential dt.

Todi
.

In fact, most often there is a problem in which replacement is made, choosing a new one as a function of the old one: t = t (x). We guessed that the integral function can be seen
,
de t′ (x)- this is not the case t x , then
.

Also, the basic formula for replacing the flour can be used in two types.
(1) ,
de x - tse function vіd t.
(2) ,
where t is a function of x.

More respectfully

In tables of integrals, changeover integration is most often denoted as x. Please note that the changeable integration can be designated by any letter. And moreover, how variable integration can be effective.

Let's look at the tabular integral as a butt
.

Here x can be replaced with any other changeable function or with a changeable function. The axis of the butt is possible:
;
;
.

In the remaining application, the following is corrected, so that when moving to variable integration x, the differential is transformed as follows:
.
Todi
.

Which application has the essence of integration by substitution. Then we can guess what
.
After this integral is reduced to the tabular one.
.

You can calculate this integral using the additional replacement formula (2) . Let's put t = x 2+x. Todi
;
;

.

Integration application by replacement

1) Computable integral
.
We note that (sin x)′ = cos x. Todi

.
Here we put in the substitution t = sin x.

2) Computable integral
.
Let's note that... Todi

.
Here we viconal integration by replacing the change t = arctan x.

3) Integrated
.
Let's note that... Todi

. Here, when integrated, it is possible to replace the variable t = x 2 + 1 .

Linear substitutions

Perhaps the most extensive are linear substitutions. This is a replacement for a changeable look
t = ax + b,
de a and b - permanent. For such replacements, differentials are associated with joint relations
.

Applications of integration with linear substitutions

A) Evaluate the integral
.
Decision.
.

B) Find the integral
.
Decision.
It is accelerated by the authorities of the display function.
.
ln 2– it’s not static. The integral can be calculated.

.

C) Evaluate the integral
.
Decision.
Let us reduce the quadratic term from the sign to the sum of squares.
.
The integral can be calculated.

.

D) Find the integral
.
Decision.
Soluble rich compound under the root.

.
Integrated, stagnant method of replacing the replacement.

.
Previously, we rejected the formula
.
Zvidsi
.
Having submitted this, we can discard the residual evidence.

Type of occupation: Development of new material.

Headquarters:

learn to establish the method of integration by substitution;
continue to mold the memory and skills of the integration of functions;
continue to formulate interest to mathematics;
ensure that you are aware of the situation before starting the process, take advantage of your knowledge, exercise self-control over the process of obtaining the right;
Remember that you are only aware of the complexity of the algorithms for calculating the unvalued integral, allowing students to clearly understand the topic that is being taught.

Carefree activities:

table of basic integration formulas;
task cards for turning robots.

The student is guilty of the nobility: algorithm for calculating the unvalued integral using the substitution method.

The student is responsible for: set up the abstraction for calculating non-valuable integrals.

Motivation for cognitive activity of students.

The calculation report informs that in addition to the method of medianless integration, there are other methods for calculating non-significant integrals, one of which is the substitution method. This is the most extensive integration method folding function, which transforms into a re-integration of the integral after an additional transition to another change integration.

Stay busy

I. Organizational moment.

II. Checking the homework.

Frontal feeding:

III. Repetition of basic knowledge of teaching.

1) Repeat the table of basic integration formulas.

2) Repeat the method of middle-free integration.

Direct integration is a method of integration in which the integral by means of the same transformation of the integral function and the consolidation of the powers of the unvalued integral is brought to one or many tabular ints egraliv.

IV. Development of new material.

The task of calculating the integral of the medianless integrations is always possible, since it is associated with great difficulties. In such situations, you need to use other methods. One of the most effective techniques is the method of substitution and replacement of variable integration. The essence of this method lies in the fact that when introducing a new change integration, it is possible to reduce the tasks of the integral to a new integral, which is very easy to take directly. If, after replacing the variable integral, it becomes simpler, then the substitution mark is achieved. Integration by the substitution method is based on the formula

Let's look at this method.

Calculation algorithmunvalued integral by substitution method:

This means that up to which table integral the given integral is applied (having first reworked the integral expression, if necessary).
Determine which part of the integral function is replaced with a new one, and write down this replacement.
Find the differentials of both parts of the record and express the differential of the old change (or a expression to replace this differential) through the differential of the new change.
Turn the replacement under the integral.
Know the rejection of the integral.
Through the war, they will vibrate the turnaround replacement, then. go to the old place. The result must be verified by differentiation.

Let's take a look.

apply it. Find integrals:

1) )4

Let's introduce the substitution:

Differentiation between jealousies, perhaps:

V. Zastosuvannaya know about the rise of standard applications.

VI. Independently stagnant knowledge, learn and learn.

Option 1

Find integrals:

Option 2

Find integrals:

VII. Keeping your bags busy.

VIII. Home improvement:

G.M. Yakovlev, part 1, §13.2, paragraph 2, No. 13.13 (1,4,5), 13.15 (1,2,3)

Replacing the penis or Here - a rich step, for example, viraz - a rich step.

Let's say we have a butt:

The method of replacing the change is stagnant. What do you think should be taken for? Right, .

Rivne comes to mind:

We carry out a complete replacement of the following:

Let's get out of the way:

Virishimo friend Rivnyanya:

...What does this mean? Right! There is no solution yet.

In this manner, we removed two types -; .

Do you understand how to establish a method for replacing a substitute with a rich member? Try to do something like this on your own:

Virishiv? Now we’ll review the main points with you.

Take it for what it's worth.

We otrimuemo viraz:

Most squarely equal, we take away that there are two roots: i.

The solutions of the first square are equal to the numbers

Connections with another square level - numbers in.

Vіdpovid: ; ; ;

Let's summarize the pouches

The method of replacing the spare parts of the main types of replacement parts for equal inaccuracies:

1. Step replacement, if we accept something unknown, reduced to a step.

2. Replacement of a polynomial, if we take the whole expression, which is unknown.

3. Fractional-rational replacement, if we accept any relation in order to replace an unknown change.

Important for the sake of when introducing a new change:

1. Replacing essential items must be done immediately, whenever possible.

2. Before a new change, it is necessary to leave the rest and then return to the old unknown.

3. When turning to the cob of the unknown (that’s how it was done with this decision), don’t forget to check the root on the ODZ.

A new change must be introduced in the same way as for equalities and for inequalities.

Rozberomo 3 zavdannya

Submissions for 3 days

1. Let it go, then the look comes into view.

The fragments can be either positive or negative.

Subject:

2. Let it go, then the look comes into view.

There is no solution, fragments.

Subject:

3. Groupings are excluded from:

Let it go, then the look comes into view
.

Subject:

REPLACEMENT OF THE MINDS. MIDDLE RIVER.

Replacing spare parts- this is the introduction of a new unknown, so that jealousy and inequity take on a simpler appearance.

I will list the main types of replacements.

Step replacement

Step replacement.

For example, with the help of replacing the biquadratic, the equation is adjusted to the square: .

With nervousness everything is similar.

For example, a replacement can be made for the unevenness, and the square unevenness can be eliminated: .

Butt (virishi independently):

Decision:

This is a shot-and-rational equation (repeat), but if it is determined by the primary method (reduced to the final banner) it is not easy, as soon as we remove the equal stage, the replacement of the changeable ones will stagnate.

Everything will become much simpler after replacing: . Todi:

Now it's robimo reverse substitution:

Subject: ; .

Replacing the penis

Replacing a rich member or

Here is a rich step, then. Viraz mind

(For example, viraz is a rich step, then).

The most common replacement for the quadratic trinomial is either.

Butt:

Unleash the jealousy.

Decision:

And the replacement of replacements is once again underway.

Then I see jealousy in the future:

Root of this square row: i.

There are two types of failures. We recommend a reversible replacement for the skin:

Well, the ceremonial root is silent.

Root of this region: i.

Confirmation. .

Fractional-rational replacement

Fractional-rational replacement.

i - a rich variety of steps and clearly.

For example, with the highest turning levels, then equal to the view

Call for replacement.

I’ll show Nina how it works.

It’s easy to see that jealousy is not the root of it: even if we put it in terms of jealousy, we deny that it’s super clear to the mind.

We divide the jealousy into:

Regroupable:

Now let's quickly replace: .

Its affiliation lies in the fact that when squared, the creation of the Dodanks shortens x:

What follows.

Let's turn to our rival:

Now it is enough to square the equation and make a reverse replacement.

Butt:

Unleash the jealousy: .

Decision:

When jealousy does not end, that is. We divide the jealousy into:

I’ll see the rivalry in the future:

Yogo root:

Let's make a quick replacement:

Let's take a look at it:

Subject: ; .

More butt:

Release the nervousness.

Decision:

We reconfigure with a non-median substitution so as not to enter until the maximum value of inequality. Let us divide the number and the sign of the skin and fractions into:

Now the replacement is obvious: .

I can see that there is some nervousness in the future:

Vikorist uses the interval method for finding y:

in front of everyone, because

Well, nervousness is tantamount to attack:

in front of everyone, fragments.

Well, nervousness is tantamount to attack: .

Well, inequality appears equivalent to totality:

Subject: .

Replacing spare parts- one of the most important methods for solving rivalries and insecurities.

Finally I will give you a couple of important wishes:

REPLACEMENT OF THE MINDS. SHORT VIKLAD AND BASIC FORMULAS.

Replacing spare parts- a method of untying folded lines and irregularities, which allows you to simplify the output and bring it to a standard appearance.

Types of replacement:

Step replacement: to take on the unknown, reduced to the level - .
Fractional-rational replacement: one is embarking on an endeavor to take revenge on an unknown change - , there are many terms of steps n and m, apparently.
Replacing the penis: the purpose of expression is taken to take revenge on the unknown - abo, de - rich term step.

Once the level of uneasiness/unevenness has increased, it is necessary to carry out a reverse replacement.

Calculate the tasks of the integral without median integrations

don't go into it again. One of the most effective techniques

є method of substitution and replacement of exchange integration.

The essence of this method lies in the fact that by introducing a new change integration it is possible to know the tasks of the integral

a new integral, which can be taken without medial integrations.

Let's take a look at this method:

Let go - uninterrupted function

need to know: (1)

We are ready to replace the replacement integration:

de φ (t) - monotonic function, as there is a continuous march

This is a complex function f(φ(t)).

Having reduced the differential formula to F(x) = F(φ(t))

functions that can be disabled:

﴾F(φ(t))﴿′ = F′(x) ∙ φ′(t)

Ale F′(x) = f(x) = f(φ(t)), therefore

﴾F(φ(t))﴿′ = f(φ(t)) ∙ φ′(t) (3)

Thus, the function F(φ(t)) is primary for the function

f (φ (t)) ∙ φ′ (t), therefore:

∫ f(φ(t)) ∙ φ′(t) dt = F(φ(t)) + C (4)

Vrahovuchi, that F(φ(t)﴿ = F(x), from (1) and (4) the replacement formula follows

change for the unvalued integral:

∫ f(x)dx = ∫ f(φ(t)) φ′ (t)dt (5)

Formally, formula (5) comes out by replacing x with φ(t) and dх with φ′(t)dt

When extracted after integration with formula (5), the result follows

go again to change x. This is always possible, for whatever reason

In addition, the function x = φ (t) is monotonic.

In the future, choosing a substitution is difficult.

news For their finishing it is necessary to use the differential technology

Table integrals are valuable and good to know.

However, it is still possible to establish a number of secret rules and certain techniques

integration.

Rules for integration using the substitution method:

1. Determine to which table integral the given integral is applied (after re-creating the integral expression, if necessary).

2. Determine which part of the integral function should be replaced

new change, then write down this change.

3. Know the differentials of both parts of the record and express the differentials

the whole old change (or the opposite is true, in order to take revenge on this dif-

differential) through the differential of the new exchange.

4. Replace under the integral.

5. Know the rejection of the integral.

6. As a result, go to the old change.

Apply solving integrals using the substitution method:

1. Know: ∫ x²(3+2x) dx

Decision:

Let’s make the substitution 3+2x = t

We know the differential of both parts of the substitution:

6x dx = dt, stars

Otje:

∫ x (3+2x ) dx = ∫ t ∙ dt = ∫ t dt = ∙ + C = t + C

Replacing t with the expression from the substitution, we eliminate:

∫ x (3+2x) dx = (3+2x) + C

Decision:

= = ∫ e = e + C = e + C

Decision:

Understanding the singing integral.

The difference between the value for any primary function when changing the argument from a to is called the first integral of the function between a to b and is designated:

a and b are called the lower and upper boundaries of integration.

To calculate the integral integral, you need:

1. Find the integral of non-values

2. Place the upper part of the integration into the abdomen, and then the lower part.

3. From the first result of substitution, derive another.

Briefly, this rule can be written in formulas like this:

This formula is called the Newton–Leibniz formula.

The main powers of the singing integral:

1. , de K = const

3. Yakshcho, then

4. Since the function is not visible in the section, then

When replacing an old integral with a new one, it is necessary to replace the old interchange with a new one. These new boundaries are indicated by the opposite substitution.

Zastosuvannya of the singing integral.

The area of a curved trapezoid is surrounded by a curve, the entire abscise and two straight lines і calculated using the following formula:

The volume of a body wrapped around the abscissa axis of a curved trapezoid, surrounded by a curve that does not change its sign to , the entire abscis and two straight lines і calculated using the following formula:

With the help of the song integral, you can also control low physical tasks.

For example:

If the fluidity of a body that collapses in a straight line is a function of the hour t, then the path S that the body passes from the moment t = t 1 to the moment t = t 2 is given by the formula:

Since the changeable force is determined by the function of the path S (in which the force is transferred, but is not directly changed), then the robot A, which is affected by this force on the path, is indicated by the formula:

Apply:

1. Calculate the area of the figure surrounded by lines:

y =; y = (x-2) 2; 0x.

Decision:

a) Let's graph functions: y =; y = (x-2) 2

b) It is significant to calculate the area of each trace.

c) Significantly between integrations, connecting levels: = (x-2) 2; x = 1;

d) Calculate the area of a given figure:

S = dx + 2 dx = 1 od 2

2. Calculate the area of the figure surrounded by lines:

Y = x 2; x = y2.

Decision:

x 2 =; x 4 = x;

x (x 3 - 1) = 0

x 1 = 0; x 2 = 1

S = - x 2) dx = ( x 3\2 - ) │ 0 1 = od 2

3. Calculate the volume of the body wrapped around the 0x axis of the figure, surrounded by lines: y = ; x = 1.

Decision:

V = π dx = π) 2 dx = π = π │ = π/2 od. 3

Homemade robot control in mathematics
Variants of command.

Option No. 1

y = (x + 1) 2; y = 1 - x; 0x

Option No. 2

1. Change the three-level system in the following ways: