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 ]:
WITH geometric point From a visual perspective, 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 non-negative function y = f(x) from a geometric point of view is equal to the area of a 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 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)
In differential calculus the problem is solved: under this function ƒ(x) find its derivative(or differential). Integral calculus solves the inverse problem: find the function F(x), knowing its derivative F "(x)=ƒ(x) (or differential). The sought function F(x) is called the antiderivative of the function ƒ(x).
The function F(x) is called antiderivative function ƒ(x) on the interval (a; b), if for any x є (a; b) the equality
F " (x)=ƒ(x) (or dF(x)=ƒ(x)dx).
For example, the antiderivative of the function y = x 2, x є R, is the function, since
Obviously, any functions will also be antiderivatives
where C is a constant, since
Theorem 29. 1. If the function F(x) is an antiderivative of the function ƒ(x) on (a;b), then the set of all antiderivatives for ƒ(x) is given by the formula F(x)+C, where C is a constant number.
▲ The function F(x)+C is an antiderivative of ƒ(x).
Indeed, (F(x)+C) " =F " (x)=ƒ(x).
Let Ф(х) be some other antiderivative of the function ƒ(x), different from F(x), i.e. Ф "(x)=ƒ(х). Then for any x є (а; b) we have
And this means (see Corollary 25.1) that
where C is a constant number. Therefore, Ф(x)=F(x)+С.▼
The set of all antiderivative functions F(x)+С for ƒ(x) is called indefinite integral of the function ƒ(x) and is denoted by the symbol ∫ ƒ(x) dx.
Thus, by definition 
∫ ƒ(x)dx= F(x)+C.
Here ƒ(x) is called integrand function, ƒ(x)dx — integrand expression, X - integration variable, ∫ -sign of the indefinite integral.
The operation of finding the indefinite integral of a function is called integrating this function.
Geometrically, the indefinite integral is a family of “parallel” curves y=F(x)+C (each numerical value of C corresponds to a specific curve of the family) (see Fig. 166). The graph of each antiderivative (curve) is called integral curve.
Does every function have an indefinite integral?
There is a theorem stating that “every function continuous on (a;b) has an antiderivative on this interval,” and, consequently, an indefinite integral.
Let us note a number of properties of the indefinite integral that follow from its definition.
1. The differential of the indefinite integral is equal to the integrand, and the derivative of the indefinite integral is equal to the integrand:
d(∫ ƒ(x)dx)=ƒ(x)dх, (∫ ƒ(x)dx) " =ƒ(x).
Indeed, d(∫ ƒ(x) dx)=d(F(x)+C)=dF(x)+d(C)=F " (x) dx =ƒ(x) dx
(∫ ƒ (x) dx) " =(F(x)+C)"=F"(x)+0 =ƒ (x).
Thanks to this property, the correctness of integration is checked by differentiation. For example, equality
∫(3x 2 + 4) dx=х з +4х+С
true, since (x 3 +4x+C)"=3x 2 +4.
2. The indefinite integral of the differential of a certain function is equal to the sum of this function and an arbitrary constant:
∫dF(x)= F(x)+C.
Really,
3. The constant factor can be taken out of the integral sign:
α ≠ 0 is a constant.
Really,
(put C 1 / a = C.)
4. The indefinite integral of the algebraic sum of a finite number of continuous functions is equal to the algebraic sum of the integrals of the summands of the functions:
Let F"(x)=ƒ(x) and G"(x)=g(x). Then
where C 1 ±C 2 =C.
5. (Invariance of the integration formula).
If 
, where u=φ(x) is an arbitrary function with a continuous derivative.
▲ Let x be an independent variable, ƒ(x) be a continuous function and F(x) be its antiderivative. Then
Let us now set u=φ(x), where φ(x) is a continuously differentiable function. Consider the complex function F(u)=F(φ(x)). Due to the invariance of the form of the first differential of the function (see p. 160), we have
From here▼
Thus, the formula for the indefinite integral remains valid regardless of whether the variable of integration is the independent variable or any function of it that has a continuous derivative.
So, from the formula 
by replacing x with u (u=φ(x)) we get 
In particular,
Example 29.1. Find the integral 
where C=C1+C 2 +C 3 +C 4.
Example 29.2. Find the integral Solution:
- 29.3. Table of basic indefinite integrals
Taking advantage of the fact that integration is the inverse action of differentiation, one can obtain a table of basic integrals by inverting the corresponding formulas of differential calculus (table of differentials) and using the properties of the indefinite integral.
For example, because
d(sin u)=cos u . du
The derivation of a number of formulas in the table will be given when considering the basic methods of integration.
The integrals in the table below are called tabular. They should be known by heart. In integral calculus there are no simple and universal rules for finding antiderivatives of elementary functions, as in differential calculus. Methods for finding antiderivatives (i.e., integrating a function) are reduced to indicating techniques that bring a given (sought) integral to a tabular one. Therefore, it is necessary to know table integrals and be able to recognize them.
Note that in the table of basic integrals, the integration variable can denote both an independent variable and a function of the independent variable (according to the invariance property of the integration formula).
The validity of the formulas below can be verified by taking the differential on the right side, which will be equal to the integrand on the left side of the formula.
Let us prove, for example, the validity of formula 2. The function 1/u is defined and continuous for all values of and other than zero.
If u > 0, then ln|u|=lnu, then 
That's why
If u<0, то ln|u|=ln(-u). Но
Means
So, formula 2 is correct. Similarly, let's check formula 15:
Table of main integrals
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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 its 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 limits and derivatives, 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 our article about how to calculate derivatives.
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 the sum of an infinitely large number of 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.
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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.
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
In fact, this property is 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.
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
In fact, this property is 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.