The method of replacing the change and the particularity of its stagnation. Integration with the replacement path (substitution method). Applications of integration with linear substitutions

In this lesson, we will learn about one of the most important and most widespread techniques that is blocked in the process of increasing unimportant integrals - the way of replacing the variable. To successfully master the material, basic knowledge and integration skills are required. If it seems like an empty full kettle in the integral calculation, then firstly familiarize yourself with the material, where I explain in an accessible form that I have also integrated and clearly outline the basic butts for the beginning.

Technically, the method of replacing a variable in an unidentified integral is implemented in two ways:

- Submitted functions under the differential sign;
- Vlasna replacement of the change.

In essence, the same thing, but the design of the decision looks different.

It’s more than just a simple outburst.

Submitted functions under the differential sign

In class Non-value integral. Apply your decision We started to open the differential, I guess the butt I pointed at:

So, open the differential – this is formally the same thing that you can find out.

Butt 1

Vikonati re-verify.

Looking at the table of integrals, there is a similar formula: . But the problem lies in the fact that under the sine we have not just the letter “X”, but a complex expression. What is it timid?

Let's put the function under the differential sign:

Having opened the differential curve, it is easy to verify that:

Actually i - This is a record of one and the same.

Prote, we ran out of food, and as we thought, first of all we need to write down our integral like this: ? Why is this so, and why not otherwise?

Formula (and all other tabular formulas) fairness and stagnation are NOT ONLY for changeable, but also for any collapsible virus EXCEPT B ARGUMENT FUNCTION(- Our butt) І VIRAZ UNDER THE SIGN OF THE DIFFERENTIAL BOULIE AT THE SAME TIME .

Therefore, it is obvious that the decay at the highest level may develop approximately like this: “There is no need to increase the integral. I looked at the table and found the formula . But I have a complicated argument and I just can’t make it quickly with a formula. However, if I can be removed and under the differential sign, everything will be fine. If I write it down, then. However, the output integral does not have a multiplier-triple, so the integral function does not change unless it is multiplied by. In the course of approximately such obvious fading, the following entry is made:

Now you can use the tabular formula :

Ready

Unity, we don’t have the letter “X”, but a collapsible expression.

Let's check it out. Here is a table of similarities and differentiations:

The output integral function has been removed and the integral has been found correctly.

Please note that at the time of rechecking, we vikoristavulovaya rule of differentiation of a folding function . In essence, the functions are subsumed under the sign of differential – there are two mutually reversing rules.

Butt 2

Let's analyze the integral function. Here we have a difference, and in the sign there is a linear function (with “X” as the first world). Looking at the table of integrals, we find the closest similarity: .

Let's put the function under the differential sign:

Those who find it difficult to quickly get rid of things when they need to multiply can quickly open the differential: . Yeah, that means, so that nothing changes, I don’t need to multiply the integral by .
Below is the table formula: :

Verification:


The output integral function has been removed and the integral has been found correctly.

Butt 3

Find the non-value integral. Vikonati re-verify.

Butt 4

Find the non-value integral. Vikonati re-verify.

Tse butt for independent decision. Let's finish the lesson.

When the singing reaches the perfection of integrals, similar butts sound light and click like peas:

At the end of this paragraph, I would like to also point out the “free” option, if in a linear function a change is made from a single coefficient, for example:

Strictly speaking, the solution may look like this:

As you can see, the transfer of functions under the differential sign was “painless”, without any additions. Therefore, in practice, such long-term decisions are often difficult to write down immediately, so . If necessary, be prepared to explain the calculations how you calculated them! There are no fragments of the integral in the table.

Method for replacing a variable in an unidentified integral

Let's move on to look at the lateral addition - the method of replacing the variables in an unsigned integral.

Butt 5

Find the non-value integral.

As a butt, I took the integral that we looked at in the beginning of the lesson. As we have already said, for the vertical integral we have a tabular formula , and everyone on the right would like to lead to her.

The idea behind the replacement method is to replace the expression (or function) with one letter.
Somebody asks:
The other for the popularity of the writer for replacement is the writer.
In principle, it is possible to emulate other literatures, but still adhere to the tradition.

Otje:
When replaced, we will lose it! Singingly, whoever guessed that there is a transition to a new change, then the new integral can all be expressed through the letter , and the differential has no place there.
A logical summary of what you need transform it into a creative way that only lasts for a long time.

Diya taka. After we have chosen a replacement, in this application, we need to know the differential. Given the differentials, I guess everyone’s friendship has already been established.

Oskolki, then

After analyzing the differential, I recommend rewriting the residual result as briefly as possible:
Now, based on the rules of proportion, we determine what we need:

In the pouch:
In this manner:

And this is the table integral itself (The table of integrals, of course, is also valid for the variable).

At the end of the day, it became impossible to make a replacement. Guess what.

Ready.

The finished design of the examined butt may look something like this:

“

Let's replace:

“

The icon for a non-current mathematical sense, it means that we have interrupted the solutions for further explanations.

When decorating the buttstock, the overlay badge of the collar replacement is sewn rather than embossed with a simple olive.

Respect! In such butts, the differential differential is marked with a report and there is nothing to notice.

And now is the time to guess the first way to cherries:

What's the difference? There is no difference in principle. This is actually the same. However, from the looks of the design, the method of bringing the function under the differential sign is very short..

It's the food that's to blame. Since the first method is short, then should we choose the replacement method? On the right, for low integrals it is not so easy to “adapt” the function to the differential sign.

Butt 6

Find the non-value integral.

Let’s make a replacement: (it’s important to note the replacement here)

As you see, as a result of the replacement, the output integral is significantly reduced - rising to the original static function. This is the same method of substitution - simplify the integral.

Line people can easily calculate this integral by subscribing the function under the differential sign:

In other words, such a solution is clearly not for all students. In addition, this application already has a different method of subordinating the functions under the differential sign there is a significant risk of getting lost with the authorities.

Butt 7

Find the non-value integral. Vikonati re-verify.

Butt 8

Find the non-value integral.

Substitution:
Lost z'yasuvati, what to change

Okay, we figured out what else is there to work with “X”, what if you lost your job in numbers?!
Sometimes, in the course of solving integrals, an offensive trick arises: we can figure it out from this very replacement!

Butt 9

Find the non-value integral.

This is an example of independent decision. Let's finish the lesson.

Butt 10

Find the non-value integral.

Singingly, the deeds expressed their respect that in my ancestor’s table there is no rule for replacing the changeable one. Zrobleno tse svidomo. The rule would bring confusion to the explanation, the fragments of the squinting butts do not appear in the obvious view.

The time has come to tell about the main change of mind and the method of replacing the change: The integral expression has the same function:(functions may or may not be available)

In connection with this, when integrals are found, you often have to look at the table of related ones.

In the example, it is noted that the number level is one less than the standard level. From the table of similarities we have a formula that reduces the step by one. So, if you are designated as a signer, then there are great chances that the numberer will turn out well.

Let's move on to look at the lateral addition - the method of replacing the variables in an unsigned integral.

Butt 5

Like the butt is quite integral, which we looked at at the beginning of the lesson. As we have already said, for the vertical integral we have a tabular formula ,

And everyone on the right would like to know before her.

The idea behind the replacement method is to Replace the folding form (or function) with one letter.

Somebody asks:

A friend for the popularity of the letter for replacement - the whole letter z. In principle, it is possible to emulate other literatures, but still adhere to the tradition.

We are running out of replacements before the hour dx! Chantly, whoever guessed that the transition to a new change is going on t, then in the new integral everything can be expressed through the letter t, and differential dx there is no place there at all. The following is a logical conclusion that dx required transform it into a creative way that only lasts for a long timet.

Diya taka. After we have chosen a replacement, in this application we need to know the differential dt.

Now the rules of proportion determine dx:

.

In this manner:

.

And this is the table integral itself

(The table of integrals is natural and valid for the variable t).

At the end of the day, it became impossible to make a replacement. Guess what.

The finished design of the examined butt may look something like this:

Let's replace: then

.

.

The icon for a non-current mathematical sense, it means that we have interrupted the solutions for further explanations.

When decorating the buttstock, the overlay badge of the collar replacement is sewn rather than embossed with a simple olive.

Respect! On the front butts, a new replacement differential will not be reported.

Guess the first way to guess:

What's the difference? There is no difference in principle. This is actually the same.

However, from the looks of the design, the method of transferring the function to the differential sign is quite short.

It's the food that's to blame. Since the first method is short, then should we choose the replacement method? On the right, for low integrals it is not so easy to “adapt” the function to the differential sign.

Butt 6

Find the non-value integral.

.

Let's replace:

;

.

As you see, as a result of the replacement, the output integral is significantly reduced - rising to the original static function. This is the same method of substitution - simplify the integral.

Line people can easily calculate this integral by subscribing the function under the differential sign:

In other words, such a solution is clearly not for all students. In addition, this application already has a different method of subordinating the functions under the differential sign there is a significant risk of getting lost with the authorities.

Butt 7

Find the non-value integral

Vikonati re-verify.

Butt 8

Find the non-value integral.

.

Decision: We are replacing: .

.

Lost z'yasuvati, what to change xdx? Once the integrals are solved, the next trick appears: x We can understand this by replacing:

.

Butt 9

Find the non-value integral.

This is an example of independent decision. Let's finish the lesson.

Butt 10

Find the non-value integral.

Singingly, the deeds expressed their respect that in the antecedents’ table there is no rule for replacing a changeable one. Zrobleno tse svidomo. The rule would have brought the rogue to the clarification of that understanding, the fragments of the ghastly-looking butts do not appear in the obvious view.

The time has come to tell about the main change of mind and the method of replacing the change: the integral expression has a singing function it's cool. For example, yak : .

F functions that may be above the creation, or subordinated to others.

In connection with this, when integrals are found, you often have to look at the table of related ones.

In case 10, it is noted that the number level is one less than the standard level. From the table of similarities we have a formula that reduces the step by one. Well, what do you mean for t signifier, then there are great chances, as well as a number xdx pretend to be good:

Substitution: .

Before speaking, it is not so difficult to put the function under the differential sign:

It should be noted that for fractions such a trick will no longer work (more precisely, apparently, it will be necessary not only to use a replacement method).

You can learn how to integrate fractions in class Integration of folding shot. There are also a couple of standard applications for independent development using the same method.

Butt 11

Find the non-value integral

Butt 12

Find the non-value integral

Solutions at the end of the lesson.

Butt 13

Find the non-value integral

.

We can see in the table of similarities and find our arc cosine: , The fragments in our integrand are found to be the arccosine and it is similar to this.

Zagalne rule:

Behind t signifies the function itself(Don't go away).

In this section: . The connection has become lost, so the part of the integral expression that has been lost will be transformed.

Whose butt is overused d t Let's write a report, fragments - folding function:

Abo, in short:

.

The rule of proportion determines the surplus we need: .

In this manner:

Butt 14

Find the non-value integral.

.

An example of independent decision. The word is already close.

Dear readers have noted that we have seen few applications of trigonometric functions. And it’s not damaged, the fragments are under and integrals of trigonometric functions introduced around lessons 7.1.5, 7.1.6, 7.1.7. Moreover, some basic guidelines are given for replacing the replacement part, which is especially important for teapots, for which it is not immediately clear that the replacement itself needs to be carried out in any other integral. The same types of replacements can be found in Article 7.2.

More advanced students can learn from the standard substitution in integrals with irrational functions

Example 12: Decision:

Let's replace:

Example 14: Decision:

Let's replace:

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 variable is that from the output variable integration, never mind x, we move on to the next variable, which is significant as 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 the expression that 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 determine the integral function f (x) through the change t you just need to substitute the change x in the form 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)- tse is similar to 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, as a changeable integration, that may be the case.

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, during integration, the replacement of 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, static method of replacing the changeable.

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

When calculating the integrals from the vicistrals of the Newton-Leibniz formula, it is important not to strictly separate the stages of the related problem (finding the primary integral function, finding an increase in the primary one). This approach, which vikorista, concretizes, replaces the formulas and integrates by parts for the integral integral, therefore allows us to simplify the recording of the solution.

THEOREM. Let the function φ(t) move continuously to the cut [α,β], а=φ(α), в=φ(β), and the function f(х) is continuous at the skin point x in the form x=φ(t), where t[α,β].

Then such jealousy is justified:

This formula is called the formula for replacing the variable in the integral.

Similar to the situation before with the unvalued integral, a quick replacement of the variable allows us to simplify the integral, bringing it closer to the tabular one. In order to replace the unvalued integral, there is no need to return to the output variable integration. It is enough to know between the integration of α and β behind the new change t as a solution to the change t level φ(t)=а and φ(t)=в. In fact, when replacing a changeover, they often begin by indicating the ratio t=ψ(x) of the new change through the old one. In this case, the relationship between the integration of the variable t is: α = ψ (a), β = ψ (c).

Example 19. Calculate

Let's put t = 2's 2 . Then dt=d(2-х 2)=(2-х 2)"dx=-2xdx и xdx=-dt. If x=0, then t=2-0 2 =2, and if x=1, then t = 2-1 2 = 1. Otzhe:

Example 20. Calculate

It is possible to quickly replace the replacement. Todi i. If x=0, then t=1 and if x=5, then t=4. Once replaced, discarded.