Variable substitution method in the indefinite integral. Variable change in indefinite integral

2. Variable substitution (substitution method)

The essence of the substitution method is that, as a result of introducing a new variable, the given complicated the integral is reduced to a tabular or such, the calculation method of which is known.

Let it be required to calculate the integral. There are two substitution rules:

General rule for selection of functions
does not exist, but there are several types of integrands for which there are recommendations for selecting a function
.

Variable substitution can be applied multiple times until the result is obtained.

Example 1. Find integrals:

a)
; b)
; v)
;

G)
; e)
; e)
.

Solution.

a) Among the tabular integrals, there are no radicals of various degrees, so "I want to get rid of", first of all, from
and
... This will require replacing NS such an expression from which both roots would be easily extracted:

b) A typical example, when there is a desire to "get rid" of the exponential function
... But in this case, it is more convenient to take the entire expression in the denominator of the fraction as a new variable:

;

c) Noticing that the numerator contains the product
, which is part of the differential of the radical expression, replace the whole expression with a new variable:

;

d) Here, as in case a), one would like to get rid of the radical. But since, unlike point a), there is only one root, then we will replace it with a new variable:

e) Here, the choice of replacement is facilitated by two circumstances: on the one hand, an intuitive desire to get rid of logarithms, on the other hand, the presence of the expression , which is the differential of the function
... But as in the previous examples, it is better to include constants accompanying the logarithm in the replacement:

f) Here, as in the previous example, the intuitive desire to get rid of the cumbersome exponent in the integrand is consistent with the well-known fact:
(formula 8 of table 3). Therefore, we have:

Change of variables for some classes of functions

Let's consider some classes of functions for which certain substitutions can be recommended.

Table 4.Rational functions

Integral form	Integration method
1.1.
1.2.
1.3.	Selecting a full square:
1.4.	Recurrent formula

Transcendental functions:

1.5.
- substitution t = e x ;

1.6.
- substitution t= log a x.

Example 2. Find integrals of rational functions:

a)
; b)
;

v)
; e)
.

Solution.

a) There is no need to calculate this integral using a change of variables; here it is easier to use the summation under the differential sign:

b) Similarly, we use the summation under the differential sign:

;

c) Before us is an integral of type 1.3 of Table 4, we use the appropriate recommendations:

e) Similar to the previous example:

Example 3. Find integrals

a)
; b)
.

Solution.

b) The integrand contains the logarithm, so we will use recommendation 1.6. Only in this case it is more convenient to replace not just a function
, and the whole expression under the root is:

Table 6. Trigonometric functions (R

Integral form	Integration method
3.1.	Universal substitution , , ,
3.1.1. , if	Substitution
3.1.2. , if	Substitution .
3.1.3. . , if (i.e. there are only even powers of the functions )	Substitution
3.2.	If - odd, then see 3.1.1; if - odd, then see 3.1.2; if - even, then see 3.1.3; if - even, then use the degree reduction formulas ,
3.3. , ,	Use formulas

Example 4. Find integrals:

a)
; b)
; v)
; e)
.

Solution.

a) Here we integrate the trigonometric function. Let's apply the universal substitution (table 6, 3.1):

b) Here we also apply generic substitution:

Note that the change of variables in the considered integral had to be applied twice.

c) Calculate in the same way:

e) Consider two methods for calculating this integral.

As you can see, we got different antiderivative functions. This does not mean that one of the techniques used is giving the wrong result. The fact is that using the well-known trigonometric identities connecting the tangent of a half angle with the trigonometric functions of the total angle, we have

Thus, the found antiderivatives coincide with each other.

Example 5. Find integrals:

a)
; b)
; v)
; G)
.

Solution.

a) In this integral, one can also apply the universal substitution
, but since the cosine included in the integrand is in an even degree, it is more rational to use the recommendations of paragraph 3.1.3 of Table 6:

b) First, we bring all trigonometric functions included in the integrand to one argument:

In the resulting integral, we can apply a universal substitution, but we note that the integrand does not change sign when the signs of the sine and cosine change:

Consequently, the function has the properties specified in clause 3.1.3 of Table 6, so the most convenient substitution will be
... We have:

c) If the sign of the cosine is changed in the given integrand, then the whole function will change sign:

Hence, the integrand has the property described in Section 3.1.2. Therefore, it is rational to use the substitution
... But first, as in the previous example, we transform the integrand:

d) If the sign of the sine is changed in a given integrand, then the entire function will change sign, which means that we have the case described in clause 3.1.1 of Table 6, so the new variable must be designated the function
... But since neither the presence of the function is observed in the integrand
, nor its differential, we first transform:

Example 6. Find integrals:

a)
; b)
;

v)
G)
.

Solution.

a) This integral refers to integrals of the form 3.2 of Table 6. Since the sine is of an odd degree, then, according to the recommendations, it is convenient to replace the function
... But first, we transform the integrand:

b) This integral is of the same type as the previous one, but here the functions
and
have even degrees, so you need to apply the degree reduction formulas:
,
... We get:

c) We transform the function:

d) According to the recommendations 3.1.3 of Table 6, in this integral it is convenient to make the substitution
... We get:

Table 5.Irrational functions (R Is a rational function of its arguments)

Integral form	Integration method
	Substitution , where k – common denominator of fractions …, .
	Substitution , where k- common denominator of fractions …,
2.3.	Substitution, , where k- common denominator of exponent fractions …,
2.4.	Substitution .
2.5.	Substitution ,
2.6.	Substitution , .
2.7.	Substitution , .
2.8. (differential bin), is integrated only in three cases: a) R- integer (substitution NS = t k, where k- common denominator of fractions T and NS); b) - whole (replacement = t k, where k- fraction denominator R); v) - whole (replacement = t k, where k- fraction denominator R).

Example 7. Find integrals:

a)
; b)
; v)
.

Solution.

a) This integral can be attributed to integrals of the form 2.1, therefore, we perform the appropriate substitution. Recall that the meaning of the substitution in this case is to get rid of irrationality. And this means that the radical expression should be replaced with such a power of the new variable, from which all the roots under the integral would be extracted. In our case, it is obvious :

The integral is an incorrect rational fraction. The integration of such fractions presupposes, first of all, the selection of the whole part. So let's divide the numerator by the denominator:

Then we get
, from here

We turn to the consideration of the general case - the method of changing variables in an indefinite integral.

Example 5

As an example, I took the integral that we considered at the very beginning of the lesson. As we already said, for solving the integral we liked the tabular formula, and I would like to reduce the whole matter to it.

The idea behind the replacement method is to replace a complex expression (or some function) with one letter.
In this case, it begs:
The second most popular replacement letter is the letter.
In principle, other letters can be used, but we will still stick to tradition.

So:
But when replacing, we still have! Probably, many have guessed that if a transition is made to a new variable, then in the new integral everything should be expressed through a letter, and there is no place for the differential.
It follows a logical conclusion that you need turn into some expression that depends only on.

The action is as follows. After we have found a replacement, in this example, we need to find the differential. With differentials, I think everyone has already established friendship.

Since then

After the showdown with the differential, I recommend rewriting the final result as short as possible:
Now, according to the rules of proportion, we express the one we need:

Eventually:
Thus:

And this is already the most tabular integral ( integral table, of course, is also valid for a variable).

In conclusion, it remains to carry out the reverse replacement. Remember that.

Ready.

The final layout of the considered example should look something like this:

“

Let's replace:

“

The icon does not carry any mathematical meaning, it means that we have interrupted the solution for intermediate explanations.

When writing an example in a notebook, it is better to superscript the reverse replacement with a simple pencil.

Attention! In the following examples, finding the differential will not be described in detail.

Now is the time to remember the first solution:

The question arises. If the first way is shorter, then why use the replace method? The point is that for a number of integrals it is not so easy to "fit" the function under the sign of the differential.

Example 6

Find the indefinite integral.

Let's make a replacement: (it's hard to think of another replacement here)

As you can see, as a result of the replacement, the original integral has become much simpler - reduced to an ordinary power function. This is the purpose of the replacement - to simplify the integral.

Lazy advanced people can easily solve this integral by putting a function under the differential sign:

Another thing is that such a solution is not obvious for all students. In addition, already in this example, the use of the method of bringing a function under the differential sign significantly increases the risk of confusion in the solution.

Example 7

Find the indefinite integral. Check it out.

Example 8

Find the indefinite integral.

Replacement:
It remains to find out what will become

Okay, we expressed it, but what to do with the “x” remaining in the numerator ?!
From time to time, in the course of solving integrals, the following trick occurs: we express from the same substitution!

Example 9

Find the indefinite integral.

This is an example for a do-it-yourself solution. The answer is at the end of the lesson.

Example 10

Find the indefinite integral.

Surely some have noticed that there is no variable replacement rule in my lookup table. This was done deliberately. The rule would confuse explanation and understanding, since it does not appear explicitly in the examples above.

Now is the time to talk about the basic premise of using the variable replacement method: the integrand must contain some function and its derivative : (functions may not be in the work)

In this regard, when finding integrals, one often has to look into the table of derivatives.

In this example, note that the degree of the numerator is one less than the degree of the denominator. In the table of derivatives, we find the formula that just lowers the degree by one. And, therefore, if you designate for the denominator, then the chances are great that the numerator will turn into something good.

Replacement:

By the way, here it is not so difficult to bring the function under the differential sign:

It should be noted that for fractions like, such a trick will no longer work (more precisely, it will be necessary to apply not only the replacement technique). You can learn how to integrate some fractions in the lesson. Integration of some fractions.

Here are a couple more typical examples for an independent solution from the same opera:

Example 11

Find the indefinite integral.

Example 12

Find the indefinite integral.

Solutions at the end of the lesson.

Example 13

Find the indefinite integral.

We look at the table of derivatives and find our arccosine: ... In our integrand we have the inverse cosine and something similar to its derivative.

General rule:
Per denote the function itself(and not its derivative).

In this case: . It remains to find out what the rest of the integrand will turn into.

In this example, I will describe the finding in detail since it is a complex function.

Or shorter:
According to the rule of proportion, we express the remainder we need:

Thus:

Here it is no longer so easy to bring the function under the differential sign.

Example 14

Find the indefinite integral.

An example for an independent solution. The answer is very close.

Astute readers will have noticed that I have looked at few examples with trigonometric functions. And this is no coincidence, since under integrals of trigonometric functions a separate lesson is given. Moreover, this lesson provides some useful guidelines for changing a variable, which is especially important for dummies who do not always and do not immediately understand what kind of replacement needs to be done in a particular integral. Also, some types of replacements can be found in the article Definite integral. Examples of solutions.

More experienced students can familiarize themselves with a typical replacement in integrals with irrational functions... Replacing when integrating roots is specific, and its execution technique differs from the one we discussed in this lesson.

Wish you success!

Example 3:Solution :

Example 4:Solution :

Example 7:Solution :

Example 9:Solution :

Replacement:

Example 11:Solution :

Let's replace:

Example 12:Solution :

Let's replace:

Example 14:Solution :

Let's replace:

Integration by parts. Examples of solutions

Hello again. Today in the lesson we will learn how to integrate piece by piece. Integration by parts is one of the cornerstones of integral calculus. On the test, the exam, the student is almost always asked to solve the integrals of the following types: the simplest integral (see articleIndefinite integral. Examples of solutions ) or the integral for the change of variable (see articleVariable change method in indefinite integral ) or the integral is just on method of integration by parts.

As always, you should have at hand: Integral table and Derivatives table... If you still do not have them, then please visit the storeroom of my website: Mathematical formulas and tables... I will not tire of repeating - it is better to print everything. I will try to present all the material consistently, simply and easily, there are no special difficulties in integration by parts.

What problem does the method of integration by parts solve? The method of integration by parts solves a very important problem, it allows you to integrate some functions that are absent in the table, work functions, and in some cases - and the quotient. As we recall, there is no convenient formula: ... But there is this: - the formula for integration by parts in person. I know, I know, you are the only one - we will work with her for the whole lesson (it's already easier).

4), - inverse trigonometric functions ("arches"), "arches", multiplied by some polynomial.

Also, some fractions are taken in parts, we will also consider the corresponding examples in detail.

Integrals of logarithms

Example 1

Find the indefinite integral.

Classic. From time to time, this integral can be found in tables, but it is undesirable to use a ready-made answer, since the teacher has spring vitamin deficiency and he will swear strongly. Because the integral under consideration is by no means tabular - it is taken piece by piece. We decide:

We interrupt the solution for intermediate explanations.

We use the formula for integration by parts:

A ways to reduce integrals to tabular we have listed for you:

variable replacement method;

method of integration by parts;

Direct integration method

ways of representing indefinite integrals in terms of tabular integrals for integrals of rational fractions;

methods for representing indefinite integrals in terms of tabular integrals for integrals of irrational expressions;

ways of expressing indefinite integrals in terms of tabular integrals for trigonometric functions.

Indefinite integral of a power function

Indefinite integral of the exponential function

But the indefinite integral of the logarithm is not a tabular integral, instead the tabular is the formula:

Indefinite integrals of trigonometric functions: Integrals of sine cosine and tangent

Indefinite integrals with inverse trigonometric functions

Reduction to a tabular view or direct integration method... With the help of identical transformations of the integrand, the integral is reduced to an integral, to which the basic rules of integration are applicable and the table of basic integrals can be used.

Example

Exercise. Find the integral

Solution. We will use the properties of the integral and bring this integral to a tabular form.

Answer.

Technically variable replacement method in the indefinite integral is realized in two ways:

– Bringing a function under the differential sign. - The actual replacement of the variable.

Assigning a function under the differential sign

Example 2

Check it out.

We analyze the integrand function. Here we have a fraction, and the denominator is a linear function (with "x" in the first degree). We look at the table of integrals and find the most similar thing:.

We bring the function under the differential sign:

Those who find it difficult to immediately figure out which fraction to multiply by, can quickly reveal the differential on a draft:. Yeah, it turns out, so that nothing changes, I need to multiply the integral by. Next, we use the tabular formula:

Examination: The original integrand is obtained, which means that the integral is found correctly.

Example 5

Find the indefinite integral.

As an example, I took the integral that we considered at the very beginning of the lesson. As we already said, to solve the integral, we liked the tabular formula , and I would like to reduce the whole matter to her.

The idea behind the replacement method is to replace a complex expression (or some function) with one letter. In this case, it suggests itself: The second most popular replacement letter is the letter. In principle, other letters can be used, but we will still stick to tradition.

So: But when replacing, we still have! Probably, many have guessed that if a transition is made to a new variable, then in the new integral everything should be expressed through a letter, and there is no place for the differential. It follows a logical conclusion that you need turn into some expression that depends only on.

The action is as follows. After we have found a replacement, in this example, we need to find the differential. With differentials, I think everyone has already established friendship.

Since then

After the showdown with the differential, I recommend rewriting the final result as short as possible: Now, according to the rules of proportion, we express what we need:

Eventually: Thus: And this is already the most tabular integral (the table of integrals is naturally also valid for a variable).

In conclusion, it remains to carry out the reverse replacement. Remember that.

Ready.

The final layout of the considered example should look something like this:

“

Let's replace:

“

The icon does not carry any mathematical meaning, it means that we have interrupted the solution for intermediate explanations.

When writing an example in a notebook, it is better to superscript the reverse replacement with a simple pencil.

Attention! In the following examples, finding the differential will not be described in detail.

Now is the time to remember the first solution:

What is the difference? There is no fundamental difference. They are actually the same thing. But from the point of view of the design of the task, the method of bringing the function under the differential sign is much shorter. The question arises. If the first way is shorter, then why use the replace method? The point is that for a number of integrals it is not so easy to "fit" the function under the sign of the differential.

Integration by parts. Examples of solutions

Integrals of logarithms

Example 1

Find the indefinite integral.

We interrupt the solution for intermediate explanations.

We use the formula for integration by parts:

The formula is applied from left to right

We look at the left side:. Obviously, in our example (and in all the others that we will consider), something needs to be denoted for, and something for.

In integrals of the type under consideration foralways denoted by the logarithm.

Technically, the design of the solution is implemented as follows, in a column we write:

That is, for we denoted the logarithm, and for - the remaining part integrand expression.

Next step: find the differential:

The differential is almost the same as the derivative, how to find it, we have already analyzed in previous lessons.

Now we find the function. In order to find the function, it is necessary to integrate right side lower equality:

Now we open our solution and construct the right side of the formula:. By the way, here is a sample of a clean solution with a few notes:

The only moment in the product I immediately rearranged in places and, since it is customary to write the multiplier before the logarithm.

As you can see, the application of the formula for integration by parts, in fact, reduced our solution to two simple integrals.

Please note that in some cases right after the application of the formula, under the remaining integral, a simplification is necessarily carried out - in the example under consideration, we have reduced the integrand by "x".

Let's check. To do this, you need to take the derivative of the answer:

The original integrand is obtained, which means that the integral is solved correctly.

During the check, we used the rule of product differentiation:. And this is no coincidence.

Integration by parts formula and the formulaAre two mutually inverse rules.

Integrals of an exponent multiplied by a polynomial

General rule: per

Example 5

Find the indefinite integral.

Using a familiar algorithm, we integrate by parts:

If you have any difficulties with the integral, then you should return to the article Variable change method in indefinite integral.

The only other thing you can do is comb the answer:

But if your computing technique is not very good, then the most profitable option is to leave an answer or even

That is, the example is considered solved when the last integral is taken. It will not be a mistake, it is another matter that the teacher may ask to simplify the answer.

Integrals of trigonometric functions multiplied by a polynomial

General rule: peralways denoted by a polynomial

Example 7

Find the indefinite integral.

We integrate piece by piece:

Hmmm ... and there is nothing to comment on.

Polynomial change or. Here are the degree polynomials, for example, the expression is the degree polynomial.

Let's say we have an example:

Let's apply the variable replacement method. What do you think should be taken for? Right, .

The equation takes the form:

We make the reverse change of variables:

Let's solve the first equation:

We will solve second the equation:

… What does this mean? Right! That there are no solutions.

Thus, we received two answers -; ...

Did you understand how to use the method of changing a variable with a polynomial? Practice doing this yourself:

Decided? Now let's check the highlights with you.

For you need to take.

We get the expression:

Solving the quadratic equation, we get that it has two roots: and.

The solution to the first quadratic equation is the numbers and

The solution of the second quadratic equation is the numbers and.

Answer: ; ; ;

Let's summarize

The variable change method has the basic types of variable changes in equations and inequalities:

1. Power substitution, when we take some unknown raised to a power for us.

2. Replacement of a polynomial, when we take as an integer expression containing an unknown.

3. Fractional rational substitution, when we take as any relation containing an unknown variable.

Important advice when introducing a new variable:

1. Change of variables should be done immediately, as soon as possible.

2. The equation for a relatively new variable must be solved to the end and only then return to the old unknown.

3. When returning to the original unknown (and indeed throughout the entire solution), do not forget to check the roots for ODZ.

The new variable is introduced in the same way, both in equations and in inequalities.

Let's analyze 3 tasks

Answers to 3 problems

1. Let, then the expression takes the form.

Since, it can be both positive and negative.

Answer:

2. Let, then the expression takes the form.

there is no solution since.

Answer:

3. By grouping we get:

Let, then the expression takes the form
.

Answer:

REPLACEMENT OF VARIABLES. AVERAGE LEVEL.

Change of variables- this is the introduction of a new unknown, for which the equation or inequality has a simpler form.

I will list the main types of substitutions.

Power substitution

Power substitution.

For example, using the substitution, the biquadratic equation is reduced to the quadratic:.

In inequalities, everything is similar.

For example, in the inequality, we make a substitution, and we get the square inequality:.

Example (decide on your own):

Solution:

This is a fractional rational equation (repeat), but it is inconvenient to solve it by the usual method (reduction to a common denominator), since we will get an equation of degree, so a change of variables is applied.

Everything will become much easier after replacing:. Then:

Now we do reverse replacement:

Answer: ; ...

Polynomial change

Replacing the polynomial or.

Here is a polynomial of degree, i.e. expression like

(for example, an expression is a degree polynomial, that is).

Most often, the replacement of the square trinomial is used: or.

Example:

Solve the equation.

Solution:

And again variable substitution is used.

Then the equation will take the form:

The roots of this quadratic equation are: and.

We have two cases. Let's make a reverse replacement for each of them:

This means that this equation has no roots.

The roots of this equation are: and.

Answer. ...

Fractional rational substitution

Fractional rational replacement.

and are polynomials of degrees and, respectively.

For example, when solving recurrent equations, that is, equations of the form

usually a replacement is used.

Now I'll show you how it works.

It is easy to check what is not the root of this equation: after all, if we substitute it into the equation, we get, which contradicts the condition.

Divide the equation into:

Let's regroup:

Now we make a replacement:.

Its beauty is that when squared in the double product of terms, x is canceled:

Hence it follows that.

Let's go back to our equation:

Now it is enough to solve the quadratic equation and do the reverse replacement.

Example:

Solve the equation:.

Solution:

For, equality is not satisfied, therefore. Divide the equation into:

The equation will take the form:

Its roots:

Let's make a reverse replacement:

Let's solve the resulting equations:

Answer: ; ...

Another example:

Solve inequality.

Solution:

By direct substitution, we see that it is not included in the solution of this inequality. Divide the numerator and denominator of each of the fractions by:

Variable replacement is now obvious:.

Then the inequality takes the form:

We use the interval method to find y:

in front of everyone, since

Hence, the inequality is equivalent to the following:

in front of everyone, since.

Hence, the inequality is equivalent to the following:.

So, inequality turns out to be equivalent to the totality:

Answer: .

Change of variables- one of the most important methods for solving equations and inequalities.

Finally, I will give you a couple of important tips:

REPLACEMENT OF VARIABLES. SUMMARY AND BASIC FORMULAS.

Change of variables- a method for solving complex equations and inequalities, which allows you to simplify the original expression and bring it to a standard form.

Variable replacement types:

Power substitution: for some unknown, raised to the power -.
Fractional rational replacement: any relation containing an unknown variable is taken as - , where and are polynomials of degrees n and m, respectively.
Polynomial replacement: an integer expression containing an unknown is taken as or, where is a polynomial of degree.

After solving the simplified equation / inequality, it is necessary to make the reverse replacement.

We turn to the consideration of the general case - the method of changing variables in an indefinite integral.

Example 5

As an example, let's take the integral that we considered at the very beginning of the lesson. As we already said, to solve the integral, we liked the tabular formula ,

and I would like to reduce the whole matter to her.

The idea behind the replacement method is to replace a complex expression (or some function) with one letter.

In this case, it begs:

The second most popular replacement letter is the letter z... In principle, other letters can be used, but we will still stick to tradition.

But when replacing, we still have dx! Probably many have guessed that if a transition is made to a new variable t, then in the new integral everything should be expressed through the letter t, and the differential dx there is no place at all. It follows a logical conclusion that dx necessary turn into some expression that depends only ont.

The action is as follows. After we have found a replacement, in this example - this, we need to find the differential dt.

Now, according to the rules of proportion, we express dx:

Thus:

And this is already the most tabular integral

(the table of integrals is naturally also valid for the variable t).

In conclusion, it remains to carry out the reverse replacement. Remember that.

The final layout of the considered example should look something like this:

Let's replace:, then

The icon does not carry any mathematical meaning, it means that we have interrupted the solution for intermediate explanations.

When writing an example in a notebook, it is better to superscript the reverse replacement with a simple pencil.

Attention! In the following examples, finding the differential of a new variable will not be described in detail.

Remember the first solution:

What is the difference? There is no fundamental difference. They are actually the same thing.

But, from the point of view of the design of the task, the method of bringing the function under the differential sign is much shorter.

Example 6

Find the indefinite integral.

Let's replace:

;

Lazy advanced people can easily solve this integral by putting a function under the differential sign:

Example 7

Find the indefinite integral

Check it out.

Example 8

Find the indefinite integral.

Solution: We make a replacement:.

It remains to find out what will become xdx? From time to time, when solving integrals, the following trick occurs: x we express from the same replacement:

Example 9

Find the indefinite integral.

This is an example for a do-it-yourself solution. The answer is at the end of the lesson.

Example 10

Find the indefinite integral.

Surely some have noticed that there is no variable replacement rule in the lookup table. This was done deliberately. The rule would confuse explanation and understanding, since it does not appear explicitly in the examples above.

Now is the time to talk about the basic premise of using the variable replacement method: the integrand must contain some function and its derivative... For example how : .

F functions may not be in the work, but in a different combination.

In this regard, when finding integrals, one often has to look into the table of derivatives.

In this Example 10, note that the degree of the numerator is one less than the degree of the denominator. In the table of derivatives, we find the formula that just lowers the degree by one. So, if we denote by t denominator, then chances are good that the numerator xdx turns into something good:

Replacement: .

By the way, here it is not so difficult to bring the function under the differential sign:

It should be noted that for fractions like, such a trick will no longer work (more precisely, it will be necessary to apply not only the replacement technique).

You can learn how to integrate some fractions in the lesson. Integration of compound fractions... Here are a couple of typical examples for an independent solution to the same method.

Example 11

Find the indefinite integral

Example 12

Find the indefinite integral

Solutions at the end of the lesson.

Example 13

Find the indefinite integral

We look at the table of derivatives and find our arccosine: , since we have the inverse cosine and something similar to its derivative in the integrand.

General rule:

Per t denote the function itself(and not its derivative).

In this case: . It remains to find out what the rest of the integrand will turn into

In this example, finding d t we will write in detail, since it is a complex function:

Or, in short:

According to the rule of proportion, we express the remainder we need: .

Thus:

Example 14

Find the indefinite integral.

An example for an independent solution. The answer is very close.

Astute readers will have noticed that we have looked at few examples with trigonometric functions. And this is not accidental, since under and Integrals of trigonometric functions separate lessons are allocated 7.1.5, 7.1.6, 7.1.7. Moreover, some useful guidelines for changing a variable are given below, which is especially important for dummies who do not always and do not immediately understand what kind of replacement needs to be done in a particular integral. Also, some types of substitutions can be found in article 7.2.

More experienced students can familiarize themselves with a typical replacement in integrals with irrational functions

Example 12: Solution:

Let's replace:

Example 14: Solution:

Let's replace: