Uncertain integral. Detailed examples of solutions. Undefined integral online

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 it still means 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.

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.

There is an overview of the methods for calculating uncertain integrals. The main integration methods that include integration of the amount and difference, making a permanent integral sign, replace the variable, integrating in parts. Special methods and techniques for the integration of fractions, roots, trigonometric and indicative functions.

Pred-like and indefinite integral

The primitive f (x) from the function f (x) is such a function, the derivative of which is equal to F (X):
F '(x) \u003d f (x), x ∈ Δ,
Where Δ - The gap on which this equation is performed.

The totality of all the primordial is called an uncertain integral:
,
where C is a constant, independent of the variable x.

Basic formulas and integration methods

Table integrals

The ultimate goal of calculating uncertain integrals - by transformations, clarify the specified integral to the expression containing the simplest or tabular integrals.
See Table Integrals \u003e\u003e\u003e

The integration rule of the amount (difference)

Making a permanent integral sign

Let c be a constant, independent of x. Then it can be submitted for the integral sign:

Replacing the variable

Let x be a function from the variable T, x \u003d φ (t), then
.
Or vice versa, t \u003d φ (x),
.

By replacing the variable, you can not only calculate simple integrals, but also to simplify the calculation of more complex.

Integration rule in parts

Integration of fractions (rational functions)

We introduce the designation. Let p k (x), q m (x), R n (x) be denoted by the degrees k, m, n, respectively, relative to the variable x.

Consider the integral consisting of fractions of polynomials (the so-called rational function):

If k ≥ n, then you first need to highlight the whole part of the fraci:
.
The integral from the polynomial S k-n (x) is calculated by the integral table.

The integral remains:
where M.< n .
To calculate it, the integrand should be decomposed on the simplest fraction.

To do this, find the roots of the equation:
Q n (x) \u003d 0.
Using the roots obtained, you need to represent the denominator in the form of a work of the factors:
Q n (x) \u003d s (x - a) n a (x - b) n b ... (x 2 + ex + f) n e (x 2 + gx + k) n g ....
Here s is the coefficient at x n, x 2 + ex + f\u003e 0, x 2 + gx + k\u003e 0, ....

After that, decompose the fraction on the simplest:

Integrating, we obtain an expression consisting of simpler integrals.
Integrals of type

The T \u003d x - a is given to the table substation.

Consider the integral:

We transform the numerator:
.
Subject to the integrand, we obtain the expression in which two integral includes:
,
.
The first, substitution T \u003d x 2 + EX + F is given to the table.
The second, according to the formula of bringing:

Located to integral

We give its denominator to the sum of the squares:
.
Then substitution, integral

It is also provided to the table.

Integration of irrational functions

We introduce the designation. Let R (U 1, U 2, ..., U N) mean a rational function from variables U 1, U 2, ..., u n. I.e
,
where p, q is polynomials from variables U 1, U 2, ..., u n.

Linear irrationality

Consider the integrals of the form:
,
where - rational numbers, M 1, N 1, ..., m s, n s - integers.
Let N be a common denominator of the numbers R 1, ..., R s.
Then the integral comes down to the integral from the rational functions of the substitution:
.

Integrals from differential binomes

Consider the integral:
,
where m, n, p is rational numbers, a, b - valid numbers.
Such integrals are reduced to integrals from rational functions in three cases.

1) If P is an integer. The substitution x \u003d t n, where n is the total denominator of the fractions M and N.
2) If - the whole. Substitution A x N + B \u003d T M, where m is the number of numbers p.
3) If - a whole. Substitution A + B X - N \u003d T M, where M is the denominator of the number P.

If none of the three numbers is an integer, then according to the Chebyshev theorem, the integrals of this species cannot be expressed by the final combination of elementary functions.

In some cases, first it is useful to bring the integral to more convenient M and P values. This can be done using formulas:
;
.

Integrals containing square root of square three

Here we consider the integrals of the form:
,

Euler substitutions

Such integrals can be reduced to integrals from rational functions of one of the three substitutions of Euler:
, with a\u003e 0;
, with C\u003e 0;
where x 1 is the root of the equation A x 2 + b x + c \u003d 0. If this equation has valid roots.

Trigonometric and hyperbolic substitutions

Direct methods

In most cases, the substitutions of the Euler lead to longer calculations than direct methods. With direct methods, the integral is given to one of the species listed below.

I type

The integral of the form:
,
where p n (x) is a polynomial degree n.

Such integrals are the method of uncertain coefficients using identity:

Differentiating this equation and equating the left and right parts, we find the coefficients a i.

II type

The integral of the form:
,
where P M (x) is a polynomial degree m.

Substitution T \u003d. (X - α) -1 This integral is driven to the previous type. If m ≥ n, then the fraction should be allocated to the whole part.

III type

The third and most complex type:
.

Here you need to make a substitution:
.
After which the integral will take the form:
.
Next, permanent α, β, you need to choose such that the coefficients at T appealed to zero:
B \u003d 0, B 1 \u003d 0.
Then the integral disintegrates the sum of the integrals of two types:
;
,
which are integrated, respectively, substitutions:
z 2 \u003d a 1 t 2 + C 1;
y 2 \u003d A 1 + C 1 T -2.

General

Integration of transcendental (trigonometric and indicative) functions

We note in advance that the methods that apply for trigonometric functionsAlso applicable for hyperbolic functions. For this reason, we will not consider the integration of hyperbolic functions separately.

Integrating rational trigonometric functions from COS X and SIN X

Consider the integrals from the trigonometric functions of the form:
,
where R is a rational function. This can also include tangents and catanges that should be converted through sinuses and cosines.

When integrating such functions, it is useful to keep in mind the three rules:
1) if R ( cOS X, SIN X) multiplied by -1 from the change of sign in front of one of the values cOS X. or sIN X., it is useful to identify another of them.
2) if R ( cOS X, SIN X) does not change from the change of sign simultaneously before cOS X. and sIN X., it is useful to put tG X \u003d T or cTG X \u003d T.
3) The substitution in all cases leads to an integral from rational fraction. Unfortunately, this substitution leads to longer computing than previous, if they are applicable.

Production of power functions from COS X and SIN X

Consider the integrals of the form:

If M and N are rational numbers, then one of the substitutions T \u003d sIN X. or t \u003d cOS X. The integral is reduced to the integral from the differential binoma.

If M and N are integers, the integrals are calculated by integrating in parts. At the same time, the following formulas are obtained:

;
;
;
.

Integration in parts

The use of the formula Euler

If the integrand is linearly relative to one of the functions
cOS AX. or sin AX.It is convenient to apply the Euler formula:
e IAX \u003d. cOS AX + ISIN AX (where i 2 \u003d - 1 ),
Replacing this feature on e IAX and highlighting valid (when replacing cOS AX.) or imaginary part (when replacing sin AX.) From the result obtained.

References:
N.M. Gunter, R.O. Kuzmin, Collection of tasks on higher mathematics, "Lan", 2003.