Find an uncertain integral: the start began, examples of solutions. Uncertain integral. Detailed examples of solutions

Integral calculus.

PRINTING FUNCTION.

Definition: FunctionF (x) is called a primitive functionthe functionF (x) on the segment, if at any point of this segment is true equality:

It should be noted that there may be infinitely many for the same function. They will differ from each other for some constant number.

F 1 (x) \u003d f 2 (x) + c.

Not certain integral.

Definition: Uncertain integralthe functionF (x) is called a set of primitive functions that are determined by the relation:

Record:

The condition for the existence of an indefinite integral on some segment is the continuity of the function on this segment.

Properties:

Example:

Finding the value of an indefinite integral is mainly due to the finding of a primitive function. For some functions, this is a rather complicated task. The following will be considered ways of finding uncertain integrals for basic classes of functions - rational, irrational, trigonometric, indicative, etc.

For convenience, the significance of uncertain integrals of most elementary functions are assembled into special integral tables that are sometimes very voluminous. They include the various most common combinations of functions. But most of the formulas presented in these tables are consequences of each other, so below the table of the main integrals with which you can get the values \u200b\u200bof uncertain integrals of various functions.

Integral	Value	Integral	Value

	lNSINX + C.



	lN.

Integration methods.

Consider three basic integration methods.

Direct integration.

The direct integration method is based on the assumption of a possible value of a primitive function with further verification of this value to differentiation. In general, we note that differentiation is a powerful tool for checking the results of integration.

Consider the use of this method using the example:

Requires the value of the integral . Based on a known differentiation formula
it can be concluded that the desired integral is equal
where C is a constant number. However, on the other hand
. Thus, we can finally conclude:

Note that, in contrast to differentiation, where, for finding a derivative, clear techniques and methods were used, the rules for finding a derivative, finally determining the derivative, for integration such methods are not available. If, when you find the derivative, we used, so to speak, constructive methods that, based on certain rules, led to the result, then when finding a primary one, it is necessary to fully rely on knowledge of the tables of derivatives and primitive.

As for the direct integration method, it is applicable only for some very limited classes of functions. Functions for which it is possible to find a primary very little from the go. Therefore, in most cases, the methods described below are used.

The method of substitution (replacement of variables).

Theorem: If you want to find an integral
But it is difficult to find a primitive, then by replacing x \u003d  (t) anddx \u003d  (t), DtP is:

Evidence : Differentiating the proposed equality:

Upon reviewed by the property number 2 of an indefinite integral:

f.(x.) dX = f.[  (t.)]  (t.) dt.

what, taking into account the introduced designations and is the initial assumption. Theorem is proved.

Example.Find an indefinite integral
.

We will replace t. = sINX., dt. = cosxdt..

Example.

Replacement
We get:

Below will be considered other examples of the application of the substitution method for various types of functions.

Integration in parts.

The method is based on the well-known formula of the derivative of the work:

(UV)  \u003d uv + Vu

where uiv are some functions from x.

In differential form: D (UV) \u003d UDV + VDU

Integrating, we get:
, and in accordance with the properties of an indefinite integral above:

or
;

Received the integration formula in parts, which allows the integrals of many elementary functions.

Example.

As can be seen, the sequential use of the integration formula in parts allows you to gradually simplify the function and bring the integral to the table.

Example.

It can be seen that as a result of the re-use of integration in parts, the function failed to simplify the table. However, the last resulting integral is no different from the source. Therefore, we move it into the left part of equality.

Thus, the integral is found at all without the use of integral tables.

Before considering in detail the integration methods of different classes of functions, we give a few more examples of finding uncertain integrals by bringing them to tabular.

Example.

Integrating elementary fractions.

Definition: Elementarythe fractions of the following four types are called:

I.
III.

II.
IV.

m, n- integers (m2, n2) ib 2 - 4ac<0.

The first two types of integrals from the elementary fractions are quite simply given to the table substitution T \u003d AX + B.

Consider the integration method of elementary fractions of the type III.

The integral of the fraction of the form III would be presented in the form:

Here, in general, it is shown to bring the integral of a fraction of the form IIIO to two table integrals.

Consider the application of the above formula on the examples.

Example.

Generally speaking, if three-stared AX 2 + BX + ensepassB 2 - 4AC\u003e 0, then the fraction by definition is not elementary, however, it can nevertheless integrate the method specified above.

Example.

We now consider the methods of integrating the simplest fractions of the IVTP.

First, consider the special case at m \u003d 0, n \u003d 1.

Then the integral of the view
it is possible to present in the database of a complete square in the form of a full square
. Let's make the following transformation:

The second integral entering into this equality will take in parts.

Denote:

For the source integral we get:

The resulting formula is called recurrent.If you apply itN-1 time, then the table integral will be
.

Let us return now to the integral from the elementary fraction of the IVT type of the general case.

In the resulting equality, the first integral by substitution t. = u. 2 + s.located to the table , and the recurrent formula considered above is applied to the second integral.

Despite the seeming complexity of the integration of the elementary fraction of the form IV, it is easy to use enough for fractions with a small degree n., And the versatility and the generality of the approach makes it possible a very simple implementation of this method on a computer.

Example:

Integrating rational functions.

Integrating rational fractions.

In order to integrate the rational fraction, it is necessary to decompose it on elementary fractions.

Theorem: If a
- the correct rational fraction, the denominator (x) of which is represented as a product of linear and quadratic multipliers (we note that any polynomial with valid coefficients can be represented in this form: P.(x.) = (x. - a.)  …(x. - b.)  (x. 2 + px. + q.)  …(x. 2 + rX. + s.)  ), then this fraction can be decomposed on the elementary following scheme:

where A I, B I, M I, N i, R I, S I are some permanent values.

In the integration of rational fractions, it is resorted to the decomposition of the initial fraction on the elementary. To find the magnitude of I, B I, M I, N i, R I, S I, use the so-called method of uncertain coefficientsThe essence of which is that in order for two polynomials to be identically equal, it is necessary and enough to be equal to the coefficients with the same degrees x.

Application of this method Consider on a specific example.

Example.

When leading to a common denominator and equating the corresponding numerals, we get:

Example.

Because The fraction is wrong, then it should be pre-highlight the whole part:

6x 5 - 8x 4 - 25x 3 + 20x 2 - 76x- 7 3x 3 - 4x 2 - 17x + 6

6x 5 - 8x 4 - 34x 3 + 12x 2 2X 2 + 3

9x 3 + 8x 2 - 76x - 7

9x 3 - 12x 2 - 51x +18

20x 2 - 25x - 25

Spread the denominator of the resulting fraction on multipliers. It can be seen that at x \u003d 3 denominator the fraci turns into zero. Then:

3x 3 - 4x 2 - 17x + 6x- 3

3X 3 - 9X 2 3X 2 + 5X- 2

Thus, 3x 3 - 4x 2 - 17x + 6 \u003d (x- 3) (3x 2 + 5x- 2) \u003d (x- 3) (x + 2) (3x-1). Then:

In order to avoid when you find undefined disclosure coefficients, grouping and solving a system of equations (which in some cases it may be quite large) used so-called method of arbitrary values. The essence of the method is that the expression obtained above are alternately somewhat (according to the number of uncertain coefficients) of arbitrary values \u200b\u200bx. To simplify calculations, it is accepted as arbitrary values \u200b\u200bto take points in which the denomoter is zero, i.e. In our case - 3, -2, 1/3. We get:

We finally get:

Example.

Find uncertain coefficients:

Then the value of the specified integral:

Integrating some trigonometrics

functions.

Integrals from trigonometric functions may be infinitely a lot. Most of these integrals cannot be calculated analytically, so consider some the main types functions that can always be integrated.

Integral View
.

Here, the R - the designation of some rational function from the variablesSinxAcosx.

Integrals of this species are calculated by substitution
. This substitution allows you to convert trigonometric function to rational.

Then

In this way:

The transformation described above is called universal trigonometric substitution.

Example.

The undoubted advantage of this substitution is that it is always possible to convert trigonometric function to rational and calculate the corresponding integral. The disadvantages include the fact that when converting it may turn out a rather complicated rational function, the integration of which will take a lot of time and strength.

However, if it is impossible to apply a more rational replacement of the variable, this method is the only one intensive.

Example.

Integral View
if a

functionR.cOSX.

Despite the possibility of calculating such an integral with a universal trigonometric substitution, more rational to apply substitution t. = sINX..

Function
it may contain as many in even degrees, and, therefore, it can be converted to a rational function of relativesinx.

Example.

Generally speaking, for the use of this method, only the oddness of the function relative to the cosine is needed, and the degree of sine, which is included in the function can be any, both in both the fractional.

Integral View
if a

functionR. is odd aboutsINX..

By analogy with the case discussed above, substitution t. = cOSX.

Example.

Integral View

functionR. even aboutsINX. andcOSX.

For converting the RV function, the substitution is used

t \u003d TGX.

Example.

Integral works of sinuses and cosine

different arguments.

Depending on the type of product, one of three formulas will be applied:

Example.

Sometimes, when integrating trigonometric functions, it is convenient to use well-known trigonometric formulas to reduce the order of functions.

Example.

Sometimes some non-standard techniques are applied.

Example.

Integrating some irrational functions.

Not every irrational function may have an integral expressed by elementary functions. To find the integral from the irrational function, apply a substitution that will allow converting a function to rational, the integral of which can always be found.

Consider some techniques for integrating various types of irrational functions.

Integral View
wheren.- natural number.

With the help of substitution
the function is rationalized.

Example.

If the composition of the irrational function includes the roots of various degrees, then as a new variable, rationally take the root of the degree equal to the smallest total multiple degrees of the roots included in the expression.

We will illustrate this on the example.

Example.

Integrating binomine differentials.

Definition: Bininominal differentialcalled expression

x. m. (a. + bX. n. ) p. dX

where m., n., and p.- Rational numbers.

As proved by Academician Chebyshev P.L. (1821-1894), the integral from the binomine differential can be expressed through elementary functions only in the following three cases:

If a r- an integer, then the integral is rationalized by substitution

where - a common denominator m.and n..

The solution of integrals is the task is light, but only for the elect. This article is for those who want to learn to understand the integrals, but does not know anything about them or almost nothing. Integral ... why is it needed? How to calculate it? What is a certain and indefinite integral? If the only integral application known to you is to get a crochet in the form of an integral icon. Something useful from hard to reach places, then welcome! Learn how to solve the integrals and why without it it is impossible to do.

We study the concept of "integral"

Integration was known in ancient Egypt. Of course, not in modern video, but still. Since then, mathematics wrote a lot of books on this topic. Especially distinguished Newton and Leibnits But the essence of things has not changed. How to understand integrals from scratch? In no way! To understand this topic, the basic knowledge of the foundations of mathematical analysis will still need. It is these fundamental information about you will find in our blog.

Uncertain integral

Let us have some kind of function f (X) .

Uncertain integral function f (X) This feature is called F (X) , the derivative of which is equal to the function f (X) .

In other words, the integral is a derivative on the contrary or primitive. By the way, about how to read in our article.

Predictive exists for all continuous functions. Also, the constant sign is often added to the primary, as the derivatives differ in the constant coincide. The process of finding the integral is called integration.

Simple example:

To constantly not to calculate the primitive elementary functions, it is convenient to reduce the table and use the ready-made values:

Certain integral

Having a deal with the concept of integral, we are dealing with infinitely small values. The integral will help calculate the figure of the figure, the mass of the inhomogeneous body, passed under the uneven movement path and much more. It should be remembered that the integral is the amount of infinitely large number Infinitely small terms.

As an example, imagine a schedule of some function. How to find an area of \u200b\u200bfigures limited by a graph of the function?

With the help of the integral! We divide the curvilinear trapezium, limited by the coordinate axes and the graph of the function, on infinitely small segments. Thus, the figure will be divided into thin columns. The sum of the area of \u200b\u200bthe columns will be the area of \u200b\u200bthe trapezoid. But remember that such a calculation will give an exemplary result. However, the smaller the segments will already be, the more accurate will be the calculation. If we reduce them to such an extent that the length will strive for zero, the amount of segments will strive for the area of \u200b\u200bthe figure. This is a specific integral that is written as follows:

Points A and B are called integration limits.

Baria Alibasov and the Group "Integral"

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Rules for calculating integrals for dummies

Properties of an uncertain integral

How to solve an indefinite integral? Here we will consider the properties of an uncertain integral, which will be useful when solving examples.

The derivative of the integral is equal to the integrand function:

The constant can be made from the sign of the integral:

The integral from the amount is equal to the amount of integrals. Also also for difference:

Properties of a specific integral

Linearity:

The integral sign changes if the integration limits are swapped:

For any Points a., b. and from:

We have already found out that a certain integral is the limit of the amount. But how to get a specific value when solving the example? For this, there is a Newton-Leibnic formula:

Examples of solutions of integrals

Below will consider several examples of finding uncertain integrals. We suggest you independently understand the subtleties of the solution, and if something is incomprehensible, ask questions in the comments.

To secure the material, see the video about how integrals are solved in practice. Do not despair if the integral is not given immediately. Ask, and they will tell you about calculating the integrals all that know themselves. With our help, any triple or curvilinear integral on a closed surface will become forces.

Find an indefinite integral (many primary or "anti-derivative") means restore the function according to a known derivative of this function. Restored multiplicate F.(x.) + FROM For function f.(x.) takes into account the integration constant C.. By the speed of movement of the material point (derivative), the law of movement of this point (primitive) can be restored; By accelerating the movement of the point - its speed and the law of movement. As can be seen, integration is a wide field for the activities of Sherlock Holmes from physics. Yes, and in the economy, many concepts are represented through the functions and their derivatives and therefore, for example, it is possible to restore product volume in a certain point in time (derivative) to restore the amount of products issued at the appropriate time.

To find an indefinite integral, a fairly small number of basic integration formulas is required. But the process of its location is much more difficult than the application of these formulas. All complexity refers not to integration, but to bring the integrated expression to this species that makes it possible to find an indefinite integral on the above-mentioned formulas mentioned above. This means that in order to start the integration practice, you need to activate the expression conversion skills obtained in high school.

Learn to find integrals we will use properties and table of uncertain integrals From the lesson on the basic concepts of this topic (opens in a new window).

There are several methods for finding an integral, of which method of replacement of the variable and integration method in parts - Mandatory gentleman's set of everyone who successfully passed the highest mathematics. However, to start mastering integration is more useful and more pleasant with the use of a decomposition method based on the following two theorems on the properties of an indefinite integral, which are for ease of referring here.

Theorem 3.A permanent multiplier in the integrand can be made for a sign of an indefinite integral, i.e.

Theorem 4.The indefinite integral of the algebraic amount of the finite number of functions is equal to the algebraic sum of the indefinite integrals of these functions, i.e.

(2)

In addition, the following rule can be useful in integration: if the expression of the integrand function contains a permanent multiplier, then the expression of the primitive is dominated by the number, reverse the constant factor, that is

(3)

Since this lesson is introduced into solving the tasks of integration, it is important to note two things that either already at the very initial stageOr somewhat later they may surprise you. Surprise due to the fact that integration - the inverse differentiation operation and an uncertain integral can be rightly called "anti-derivative".

The first thing that should not be surprised at integration. In the integral table there are formulas that have no analogues among the formulas of the derivative table . These are the following formulas:

However, it is possible to make sure that the derivatives of the expressions in the right parts of these formulas coincide with the corresponding integrated functions.

The second thing that should not be surprised at integration. Although the derivative of any elementary function is also an elementary function, undefined integrals from some elementary functions are no longer elementary functions. . Examples of such integrals may be the following:

For the development of integration techniques, the following skills will be used: Reduction of fractions, dividing the polynomial in the fractional numerator on a single-wing in the denominator (to obtain the amount of indefinite integrals), the conversion of roots to a degree, multiplication is unobed to a polynomial, the extermination. These skills are needed for the transformation of the integrand, as a result of which the amount of integrals present in the integral table should be obtained.

We find indefinite integrals together

Example 1.Find an uncertain integral

Decision. We see in the denominator of the integrand expression of the polynomial in which X is in the square. This is an almost faithful sign that you can apply a table integral 21 (with Arctangent as a result). We carry out a twice multiplier from the denominator (there is a property of the integral - a permanent multiplier can be taken out of the integral sign, above it was mentioned as theorem 3). The result of all this:

Now in the denominator the sum of the squares, which means that we can apply the mentioned tabular integral. Finally get the answer:

Example 2.Find an uncertain integral

Decision. We again apply the theorem 3 - the property of the integral, on the basis of which the constant multiplier can be made for the integral sign:

We use the formula 7 from the integral table (variable to degree) to the integrand function:

We reduce the resulting fractions and before us the end answer:

Example 3.Find an uncertain integral

Decision. Using first theorem 4, and then theorem 3 on properties, we will find this integral as the sum of three integrals:

All three integral received - tabular. We use formula (7) from the integral table with n. = 1/2, n. \u003d 2 I. n. \u003d 1/5, and then

combines all three arbitrary constants that were introduced when three integrals are located. Therefore, in similar situations, only one arbitrary permanent (constant) integration should be administered.

Example 4.Find an uncertain integral

Decision. When in a denominator of the integrated fraction - unrochene, we can minimize the numerator to the denominator. The initial integral has become two integrals:

To apply a table integral, we transform the roots to the degree and now the final answer is:

We continue to find indefinite integrals together

Example 7.Find an uncertain integral

Decision. If we transform a reactive function, erecting twisted into a square and dividing the numerator to the denominator, the initial integral will become the sum of three integrals.

The process of solving integrals in science under the name "mathematics" is called integration. With the help of integration you can find some physical quantities: Area, volume, body weight and much more.

Integrals are uncertain and defined. Consider the type of a specific integral and try to understand its physical meaning. It seems in this form: $$ \\ int ^ a _b f (x) dx $$. Distinctive trait Writing a specific integral from the uncertain in the fact that there are integration limits a and b. Now we will find out what they need, and that still means a certain integral. In the geometrical sense, such an integral equal to Square Figures bounded by curve F (x), lines A and B, and the axis oh.

Figure 1 shows that a specific integral is the same area that is painted gray. Let's check it on the simplest example. We will find the area of \u200b\u200bthe figure in the image below by the integration, and then calculate it in the usual way to multiply the length of the width.

Fig. 2 shows that $ y \u003d f (x) \u003d $ 3, $ a \u003d 1, b \u003d $ 2. Now we substitute them into the definition of the integral, we get that $$ s \u003d \\ int _a ^ bf (x) dx \u003d \\ int _1 ^ 2 3 dx \u003d $$$$ \u003d (3x) \\ Big | _1 ^ 2 \u003d (3 \\ In our case, length \u003d 3, the width of the figure \u003d 1. $$ S \u003d \\ Text (length) \\ Cdot \\ Text (width) \u003d 3 \\ CDot 1 \u003d 3 \\ Text (UR) ^ 2 $$ As you can see, everything perfectly coincided .

The question appears: how to solve the integrals are uncertain and what is the meaning? The solution of such integrals is the finding of primitive functions. This process is the opposite to find the derivative. In order to find the primary one, you can use our help in solving problems in mathematics or you need to independently unmistakably drive the properties of the integrals and the integration table of the simplest elementary functions. Finding is so $$ \\ int f (x) dx \u003d f (x) + C \\ Text (where) F (x) $ is a primitive $ F (x), C \u003d const $.

To solve the integral, you need to integrate the function $ f (x) $ via variable. If the function is a table, then the answer is recorded suitable video. If not, the process is reduced to obtaining a tabular function from the function $ f (x) $ by cunning mathematical transformations. For this is various methods and properties that consider further.

So, now make an algorithm how to solve integrals for dummies?

Algorithm for calculating integrals

We learn a certain integral or not.
If uncertain then you need to find pRINTING FUNCTION $ F (x) $ from the integrated $ f (x) $ with mathematical transformations leading to a table form $ f (x) $.
If defined, then you need to perform step 2, and then substitute the limits of $ A $ and $ b $ into the primitive function $ f (x) $. What formula is to do this in the article "Newton's Formula Leibnitsa".

Examples of solutions

So, you learned how to solve integrals for dummies, examples of solving integrals disassembled the shelves. They learned physical and geometric meaning. The decision methods will be set out in other articles.