Definite and indefinite integrals message. Integrals for dummies: how to solve, calculation rules, explanation

Let the function y = f(x) is defined on the interval [ a, b ], a < b. Let's perform the following operations:

1) let's split [ a, b] dots a = x 0 < x 1 < ... < x i- 1 < x i < ... < x n = b on n partial segments [ x 0 , x 1 ], [x 1 , x 2 ], ..., [x i- 1 , x i ], ..., [x n- 1 , x n ];

2) in each of the partial segments [ x i- 1 , x i ], i = 1, 2, ... n, choose an arbitrary point and calculate the value of the function at this point: f(z i ) ;

3) find the works f(z i ) · Δ x i , where is the length of the partial segment [ x i- 1 , x i ], i = 1, 2, ... n;

4) let's make up integral sum functions y = f(x) on the segment [ a, b ]:

From a geometric point of view, this sum σ is the sum of the areas of rectangles whose bases are partial segments [ x 0 , x 1 ], [x 1 , x 2 ], ..., [x i- 1 , x i ], ..., [x n- 1 , x n ], and the heights are equal f(z 1 ) , f(z 2 ), ..., f(z n) accordingly (Fig. 1). Let us denote by λ length of the longest partial segment:

5) find the limit of the integral sum when λ → 0.

Definition. If there is a finite limit of the integral sum (1) and it does not depend on the method of partitioning the segment [ a, b] to partial segments, nor from the selection of points z i in them, then this limit is called definite integral from function y = f(x) on the segment [ a, b] and is denoted

Thus,

In this case the function f(x) is called integrable on [ a, b]. Numbers a And b are called the lower and upper limits of integration, respectively, f(x) – integrand function, f(x ) dx– integrand expression, x– integration variable; line segment [ a, b] is called the integration interval.

Theorem 1. If the function y = f(x) is continuous on the interval [ a, b], then it is integrable on this interval.

The definite integral with the same limits of integration is equal to zero:

If a > b, then, by definition, we assume

2. Geometric meaning of the definite integral

Let on the segment [ a, b] a continuous non-negative function is specified y = f(x ) . Curvilinear trapezoid is a figure bounded above by the graph of a function y = f(x), from below - along the Ox axis, to the left and right - straight lines x = a And x = b(Fig. 2).

Definite integral of a non-negative function y = f(x) from a geometric point of view equal to area curvilinear trapezoid bounded above by the graph of the function y = f(x) , left and right – line segments x = a And x = b, from below - a segment of the Ox axis.

3. Basic properties of the definite integral

1. The value of the definite integral does not depend on the designation of the integration variable:

2. The constant factor can be taken out of the sign of the definite integral:

3. The definite integral of the algebraic sum of two functions is equal to the algebraic sum of the definite integrals of these functions:

4.If function y = f(x) is integrable on [ a, b] And a < b < c, That

5. (mean value theorem). If the function y = f(x) is continuous on the interval [ a, b], then on this segment there is a point such that

4. Newton–Leibniz formula

Theorem 2. If the function y = f(x) is continuous on the interval [ a, b] And F(x) is any of its antiderivatives on this segment, then the following formula is valid:

which is called Newton–Leibniz formula. Difference F(b) - F(a) is usually written as follows:

where the symbol is called a double wildcard.

Thus, formula (2) can be written as:

Example 1. Calculate integral

Solution. For the integrand f(x ) = x 2 an arbitrary antiderivative has the form

Since any antiderivative can be used in the Newton-Leibniz formula, to calculate the integral we take the antiderivative that has the simplest form:

5. Change of variable in a definite integral

Theorem 3. Let the function y = f(x) is continuous on the interval [ a, b]. If:

1) function x = φ ( t) and its derivative φ "( t) are continuous for ;

2) a set of function values x = φ ( t) for is the segment [ a, b ];

3) φ ( a) = a, φ ( b) = b, then the formula is valid

which is called formula for changing a variable in a definite integral .

Unlike the indefinite integral, in this case not necessary to return to the original integration variable - it is enough just to find new limits of integration α and β (for this you need to solve for the variable t equations φ ( t) = a and φ ( t) = b).

Instead of substitution x = φ ( t) you can use substitution t = g(x) . In this case, finding new limits of integration over a variable t simplifies: α = g(a) , β = g(b) .

Example 2. Calculate integral

Solution. Let's introduce a new variable using the formula. By squaring both sides of the equality, we get 1 + x = t 2 , where x = t 2 - 1, dx = (t 2 - 1)"dt= 2tdt. We find new limits of integration. To do this, let’s substitute the old limits into the formula x = 3 and x = 8. We get: , from where t= 2 and α = 2; , where t= 3 and β = 3. So,

Example 3. Calculate

Solution. Let u= log x, Then , v = x. According to formula (4)

Antiderivative function and indefinite integral

Fact 1. Integration is the inverse action of differentiation, namely, restoring a function from the known derivative of this function. The function thus restored F(x) is called antiderivative for function f(x).

Definition 1. Function F(x f(x) on some interval X, if for all values x from this interval the equality holds F "(x)=f(x), that is, this function f(x) is the derivative of the antiderivative function F(x). .

For example, the function F(x) = sin x is an antiderivative of the function f(x) = cos x on the entire number line, since for any value of x (sin x)" = (cos x) .

Definition 2. Indefinite integral of a function f(x) is the set of all its antiderivatives. In this case, the notation is used

∫

f(x)dx

,

where is the sign ∫ called the integral sign, the function f(x) – integrand function, and f(x)dx – integrand expression.

Thus, if F(x) – some antiderivative for f(x) , That

∫

f(x)dx = F(x) +C

Where C - arbitrary constant (constant).

To understand the meaning of the set of antiderivatives of a function as an indefinite integral, the following analogy is appropriate. Let there be a door (traditional wooden door). Its function is to “be a door.” What is the door made of? Made of wood. This means that the set of antiderivatives of the integrand of the function “to be a door”, that is, its indefinite integral, is the function “to be a tree + C”, where C is a constant, which in this context can denote, for example, the type of tree. Just as a door is made from wood using some tools, a derivative of a function is “made” from an antiderivative function using formulas we learned while studying the derivative .

Then the table of functions of common objects and their corresponding antiderivatives (“to be a door” - “to be a tree”, “to be a spoon” - “to be metal”, etc.) is similar to the table of basic indefinite integrals, which will be given below. The table of indefinite integrals lists common functions with an indication of the antiderivatives from which these functions are “made”. In part of the problems on finding the indefinite integral, integrands are given that can be integrated directly without much effort, that is, using the table of indefinite integrals. In more complex problems, the integrand must first be transformed so that table integrals can be used.

Fact 2. When restoring a function as an antiderivative, we must take into account an arbitrary constant (constant) C, and in order not to write a list of antiderivatives with various constants from 1 to infinity, you need to write a set of antiderivatives with an arbitrary constant C, for example, like this: 5 x³+C. So, an arbitrary constant (constant) is included in the expression of the antiderivative, since the antiderivative can be a function, for example, 5 x³+4 or 5 x³+3 and when differentiated, 4 or 3, or any other constant goes to zero.

Let us pose the integration problem: for this function f(x) find such a function F(x), whose derivative equal to f(x).

Example 1. Find the set of antiderivatives of a function

Solution. For this function, the antiderivative is the function

Function F(x) is called an antiderivative for the function f(x), if the derivative F(x) is equal to f(x), or, which is the same thing, differential F(x) is equal f(x) dx, i.e.

(2)

Therefore, the function is an antiderivative of the function. However, it is not the only antiderivative for . They also serve as functions

Where WITH– arbitrary constant. This can be verified by differentiation.

Thus, if there is one antiderivative for a function, then for it there is an infinite number of antiderivatives that differ by a constant term. All antiderivatives for a function are written in the above form. This follows from the following theorem.

Theorem (formal statement of fact 2). If F(x) – antiderivative for the function f(x) on some interval X, then any other antiderivative for f(x) on the same interval can be represented in the form F(x) + C, Where WITH– arbitrary constant.

In the next example, we turn to the table of integrals, which will be given in paragraph 3, after the properties of the indefinite integral. We do this before reading the entire table so that the essence of the above is clear. And after the table and properties, we will use them in their entirety during integration.

Example 2. Find sets of antiderivative functions:

Solution. We find sets of antiderivative functions from which these functions are “made”. When mentioning formulas from the table of integrals, for now just accept that there are such formulas there, and we will study the table of indefinite integrals itself a little further.

1) Applying formula (7) from the table of integrals for n= 3, we get

2) Using formula (10) from the table of integrals for n= 1/3, we have

3) Since

then according to formula (7) with n= -1/4 we find

It is not the function itself that is written under the integral sign. f, and its product by the differential dx. This is done primarily in order to indicate by which variable the antiderivative is sought. For example,

, ;

here in both cases the integrand is equal to , but its indefinite integrals in the cases considered turn out to be different. In the first case, this function is considered as a function of the variable x, and in the second - as a function of z .

The process of finding the indefinite integral of a function is called integrating that function.

Geometric meaning of the indefinite integral

Suppose we need to find a curve y=F(x) and we already know that the tangent of the tangent angle at each of its points is a given function f(x) abscissa of this point.

According to the geometric meaning of the derivative, the tangent of the angle of inclination of the tangent at a given point of the curve y=F(x) equal to the value of the derivative F"(x). So we need to find such a function F(x), for which F"(x)=f(x). Function required in the task F(x) is an antiderivative of f(x). The conditions of the problem are satisfied not by one curve, but by a family of curves. y=F(x)- one of these curves, and any other curve can be obtained from it by parallel translation along the axis Oy.

Let's call the graph of the antiderivative function of f(x) integral curve. If F"(x)=f(x), then the graph of the function y=F(x) there is an integral curve.

Fact 3. The indefinite integral is geometrically represented by the family of all integral curves , as in the picture below. The distance of each curve from the origin of coordinates is determined by an arbitrary integration constant C.

Properties of the indefinite integral

Fact 4. Theorem 1. The derivative of an indefinite integral is equal to the integrand, and its differential is equal to the integrand.

Fact 5. Theorem 2. Indefinite integral of the differential of a function f(x) is equal to the function f(x) up to a constant term , i.e.

(3)

Theorems 1 and 2 show that differentiation and integration are mutually inverse operations.

Fact 6. Theorem 3. The constant factor in the integrand can be taken out of the sign of the indefinite integral , i.e.

These properties are used to carry out transformations of the integral in order to reduce it to one of the elementary integrals and further calculation.

1. The derivative of the indefinite integral is equal to the integrand:

2. The differential of the indefinite integral is equal to the integrand:

3. The indefinite integral of the differential of a certain function is equal to the sum of this function and an arbitrary constant:

4. The constant factor can be taken out of the integral sign:

Moreover, a ≠ 0

5. The integral of the sum (difference) is equal to the sum (difference) of the integrals:

6. Property is a combination of properties 4 and 5:

Moreover, a ≠ 0 ˄ b ≠ 0

7. Invariance property of the indefinite integral:

If , then

8. Property:

If , then

Actually this property represents a special case of integration using the variable change method, which is discussed in more detail in the next section.

Let's look at an example:

First we applied property 5, then property 4, then we used the table of antiderivatives and got the result.

The algorithm of our online integral calculator supports all the properties listed above and will easily find a detailed solution for your integral.

Solving integrals is an easy task, but only for a select few. This article is for those who want to learn to understand integrals, but know nothing or almost nothing about them. Integral... Why is it needed? How to calculate it? What are definite and indefinite integrals?

If the only use you know of for an integral is to use a crochet hook shaped like an integral icon to get something useful out of hard-to-reach places, then welcome! Find out how to solve the simplest and other integrals and why you can’t do without it in mathematics.

We study the concept « integral »

Integration was known back in Ancient Egypt. Of course not in modern form, but still. Since then, mathematicians have written many books on this topic. Especially distinguished themselves Newton And Leibniz , but the essence of things has not changed.

How to understand integrals from scratch? No way! To understand this topic, you will still need a basic knowledge of the basics of mathematical analysis. We already have information about , necessary for understanding integrals, on our blog.

Indefinite integral

Let us have some function f(x) .

Indefinite integral function f(x) this function is called F(x) , whose derivative is equal to the function f(x) .

In other words, an integral is a derivative in reverse or an antiderivative. By the way, read about how in our article.

An antiderivative exists for all continuous functions. Also, a constant sign is often added to the antiderivative, since the derivatives of functions that differ by a constant coincide. The process of finding the integral is called integration.

Simple example:

In order not to constantly calculate antiderivatives of elementary functions, it is convenient to put them in a table and use ready-made values.

Complete table of integrals for students

Definite integral

When dealing with the concept of an integral, we are dealing with infinitesimal quantities. The integral will help to calculate the area of ​​a figure, the mass of a non-uniform body, the distance traveled during uneven movement, and much more. It should be remembered that an integral is an infinite sum large quantity infinitesimal terms.

As an example, imagine a graph of some function.

How to find the area of ​​a figure bounded by the graph of a function? Using an integral! Let us divide the curvilinear trapezoid, limited by the coordinate axes and the graph of the function, into infinitesimal segments. This way the figure will be divided into thin columns. The sum of the areas of the columns will be the area of ​​the trapezoid. But remember that such a calculation will give an approximate result. However, the smaller and narrower the segments, the more accurate the calculation will be. If we reduce them to such an extent that the length tends to zero, then the sum of the areas of the segments will tend to the area of ​​the figure. This is a definite integral, which is written like this:

Points a and b are called limits of integration.

« Integral »

By the way! For our readers there is now a 10% discount on

Rules for calculating integrals for dummies

Properties of the indefinite integral

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

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

• The constant can be taken out from under the integral sign:

• The integral of the sum is equal to the sum of the integrals. This is also true for the difference:

Properties of a definite integral

• Linearity:

• The sign of the integral changes if the limits of integration are swapped:

• At any points a, b And With:

We have already found out that a definite integral is the limit of a sum. But how to get a specific value when solving an example? For this there is the Newton-Leibniz formula:

Examples of solving integrals

Below we will consider the indefinite integral and examples with solutions. We suggest you figure out the intricacies of the solution yourself, and if something is unclear, ask questions in the comments.

To reinforce the material, watch a video about how integrals are solved in practice. Don't despair if the integral is not given right away. Contact a professional service for students, and any triple or curved integral over a closed surface will be within your power.

The main task of differential calculus is to find the derivative f'(x) or differential df=f'(x)dx functions f(x). In integral calculus the inverse problem is solved. According to a given function f(x) you need to find such a function F(x), What F'(x)=f(x) or dF(x)=F'(x)dx=f(x)dx.

Thus, the main task of integral calculus is the restoration of function F(x) by the known derivative (differential) of this function. Integral calculus has numerous applications in geometry, mechanics, physics and technology. It gives a general method for finding areas, volumes, centers of gravity, etc.

Definition. FunctionF(x), , is called the antiderivative of the functionf(x) on the set X if it is differentiable for any andF'(x)=f(x) ordF(x)=f(x)dx.

Theorem. Any continuous line on the interval [a;b] functionf(x) has an antiderivative on this segmentF(x).

Theorem. IfF 1 (x) andF 2 (x) – two different antiderivatives of the same functionf(x) on the set x, then they differ from each other by a constant term, i.e.F 2 (x)=F 1x)+C, where C is a constant.

Indefinite integral, its properties.

Definition. TotalityF(x)+From all antiderivative functionsf(x) on the set X is called an indefinite integral and is denoted:

- (1)

In formula (1) f(x)dx called integrand expression,f(x) – integrand function, x – integration variable, A C – integration constant.

Let us consider the properties of the indefinite integral that follow from its definition.

1. The derivative of the indefinite integral is equal to the integrand, the differential of the indefinite integral is equal to the integrand:

And .

2. The indefinite integral of the differential of a certain function is equal to the sum of this function and an arbitrary constant:

3. The constant factor a (a≠0) can be taken out as the sign of the indefinite integral:

4. The indefinite integral of the algebraic sum of a finite number of functions is equal to the algebraic sum of the integrals of these functions:

5. IfF(x) – antiderivative of the functionf(x), then:

6 (invariance of integration formulas). Any integration formula retains its form if integration variable replace with any differentiable function of this variable:

Whereu is a differentiable function.

Table of indefinite integrals.

Let's give basic rules for integrating functions.

Let's give table of basic indefinite integrals.(Note that here, as in differential calculus, the letter u can be designated as an independent variable (u=x), and a function of the independent variable (u=u(x)).)

(n≠-1). (a >0, a≠1). (a≠0). (a≠0). (|u| > |a|).(|u|< |a|).

Integrals 1 – 17 are called tabular.

Some of the above formulas in the table of integrals, which do not have an analogue in the table of derivatives, are verified by differentiating their right-hand sides.

Change of variable and integration by parts in the indefinite integral.

Integration by substitution (variable replacement). Let it be necessary to calculate the integral

, which is not tabular. The essence of the substitution method is that in the integral the variable X replace with a variable t according to the formula x=φ(t), where dx=φ’(t)dt.

Theorem. Let the functionx=φ(t) is defined and differentiable on a certain set T and let X be the set of values ​​of this function on which the function is definedf(x). Then if on the set X the functionf(