Antiderivative and indefinite integral.
An antiderivative function f(x) on the interval (a; b) is such a function F(x) that equality holds for any x from a given interval.
If we take into account the fact that the derivative of the constant C is equal to zero, then the equality 
. Thus, the function f(x) has a set of antiderivatives F(x)+C, for an arbitrary constant C, and these antiderivatives differ from each other by an arbitrary constant value.
The whole set of antiderivatives of the function f(x) is called the indefinite integral of this function and is denoted 
.
The expression is called the integrand, and f(x) is called the integrand. The integrand is the differential of the function f(x).
The action of finding an unknown function by its given differential is called indefinite integration, because the result of integration is not one function F(x), but the set of its antiderivatives F(x)+C.
Table integrals
The simplest properties of integrals
1. The derivative of the integration result is equal to the integrand.
2. The indefinite integral of the differential of a function is equal to the sum of the function itself and an arbitrary constant.
3. The coefficient can be taken out of the sign definite integral.
4. The indefinite integral of the sum/difference of functions is equal to the sum/difference of the indefinite integrals of functions.
Intermediate equalities of the first and second properties indefinite integral are given for clarification.
To prove the third and fourth properties, it suffices to find the derivatives of the right-hand sides of the equalities:
These derivatives are equal to the integrands, which is the proof by virtue of the first property. It is also used in the last transitions.
Thus, the integration problem is the inverse problem of differentiation, and there is a very close connection between these problems:
the first property allows checking integration. To check the correctness of the integration performed, it is enough to calculate the derivative of the result obtained. If the function obtained as a result of differentiation turns out to be equal to the integrand, then this will mean that the integration has been carried out correctly;
the second property of the indefinite integral allows us to find its antiderivative from the known differential of a function. The direct calculation of indefinite integrals is based on this property.
1.4 Invariance of integration forms.
Invariant integration is a type of integration for functions whose arguments are elements of a group or points of a homogeneous space (any point of such a space can be transferred to another by a given action of the group).
function f(x) is reduced to the calculation of the integral of the differential form f.w, where
An explicit formula for r(x) is given below. The agreement condition has the form 
.
here Tg means the shift operator on X using gOG: Tgf(x)=f(g-1x). Let X=G be a topology, a group acting on itself by left shifts. I. and. exists if and only if G is locally compact (in particular, on infinite-dimensional groups, an int. does not exist). For a subset of I. and. characteristic function cA (equal to 1 on A and 0 outside A) defines the left Haar measure m(A). The defining property of this measure is its invariance under left shifts: m(g-1A)=m(A) for all gОG. The left Haar measure on a group is uniquely defined up to a set scalar factor. If the Haar measure m is known, then I. and. function f is given by the formula 
. The right Haar measure has similar properties. There is a continuous homomorphism (mapping that preserves the group property) DG of the group G into the group (with respect to multiplication) put. numbers for which
where dmr and dmi are the right and left Haar measures. The function DG(g) is called. module of the group G. If , then the group G is called. unimodular; in this case, the right and left Haar measures are the same. Compact, semisimple, and nilpotent (in particular, commutative) groups are unimodular. If G is an n-dimensional Lie group and q1,...,qn is a basis in the space of left-invariant 1-forms on G, then the left Haar measure on G is given by the n-form . In local coordinates for calculation
forms qi, you can use any matrix implementation of the group G: the matrix 1-form g-1dg is left-invariant, and its coef. are left-invariant scalar 1-forms, from which the desired basis is chosen. For example, the full matrix group GL(n, R) is unimodular and the Haar measure on it is given by a form. Let be 
X=G/H is a homogeneous space for which the locally compact group G is a transformation group and the closed subgroup H is a stabilizer of some point. In order for an I.I. to exist on X, it is necessary and sufficient that the equality DG(h)=DH(h) holds for all hОH. In particular, this is true when H is compact or semisimple. Complete theory of I. and. does not exist on infinite-dimensional manifolds.
Change of variables.
Let the function y = f(x) is defined on the interval [ a, b ], a < b. Let's perform the following operations:
1) split [ a, b] points a = x 0 < x 1 < ... < x i- 1 < x i < ... < x n = b on the 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 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) compose integral sum functions y = f(x) on the segment [ a, b ]:
With geometric point 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 f(z 1 ) , f(z 2 ), ..., f(z n) respectively (Fig. 1). Denote by λ length of the largest 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 splitting the segment [ a, b] into partial segments, nor from the choice of points z i in them, then this limit is called definite integral from function y = f(x) on the segment [ a, b] and denoted
Thus,
In this case, the function f(x) is called integrable on the [ a, b]. Numbers a and b are called the lower and upper limits of integration, respectively, f(x) is the integrand, f(x ) dx- integrand, x– integration variable; line segment [ a, b] is called the interval of integration.
Theorem 1. If the function y = f(x) is continuous on the segment [ a, b], then it is integrable on this interval.
The definite integral with the same limits of integration is equal to zero:
If a a > b, then, by definition, we set
2. The geometric meaning of a definite integral
Let on the interval [ a, b] continuous non-negative function y = f(x ) . Curvilinear trapezoid is called a figure bounded from above by the graph of a function y = f(x), from below - by the Ox axis, to the left and right - by 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 is equal to the area of a curvilinear trapezoid bounded from above by the graph of the function y = f(x) , on the left and on the right - by line segments x = a and x = b, from below - by a segment of the Ox axis.
3. Basic properties of a definite integral
1. The value of the definite integral does not depend on the notation of the integration variable:
2. A constant factor can be taken out of the sign of a 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, then
5. (mean value theorem). If the function y = f(x) is continuous on the segment [ a, b], then on this segment there exists a point such that
4. Newton–Leibniz formula
Theorem 2. If the function y = f(x) is continuous on the segment [ a, b] and F(x) is any of its antiderivatives on this segment, then the following formula is true:
which is called Newton-Leibniz formula. Difference F(b) - F(a) is written as follows:
where the character is called the double wildcard character.
Thus, formula (2) can be written as:
Example 1 Calculate Integral
Decision. 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, which has the simplest form:
5. Change of variable in a definite integral
Theorem 3. Let the function y = f(x) is continuous on the segment [ a, b]. If a:
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
which is called change of variable formula 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 integration limits α and β (for this it is necessary to solve for the variable t equations φ ( t) = a and φ ( t) = b).
Instead of substitution x = φ ( t) you can use the substitution t = g(x) . In this case, finding new limits of integration with respect to the variable t simplifies: α = g(a) , β = g(b) .
Example 2. Calculate Integral
Decision. Let's introduce a new variable according to the formula . Squaring both sides of the equation , 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, we 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
Decision. Let be u=ln x, then , v = x. By formula (4)
Antiderivative function and indefinite integral
Fact 1. Integration is the opposite of differentiation, namely, the restoration of a function from the known derivative of this function. The function restored in this way F(x) is called primitive 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 F "(x)=f(x), i.e given function f(x) is the derivative of the antiderivative function F(x). .
For example, the function F(x) = sin x is the antiderivative for 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 collection of all its antiderivatives. This uses the notation
∫
f(x)dx
,where is the sign ∫ is called the integral sign, the function f(x) is an integrand, and f(x)dx is the integrand.
Thus, if F(x) is some antiderivative for f(x) , then
∫
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 (a traditional wooden door). Its function is "to be a door". What is the door made of? From a tree. This means that the set of antiderivatives of the integrand "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, a tree species. Just as a door is made of wood with some tools, the derivative of a function is "made" of the antiderivative function with formula that we learned by studying the derivative .
Then the table of functions of common objects and their corresponding primitives ("to be a door" - "to be a tree", "to be a spoon" - "to be a metal", etc.) is similar to the table of basic indefinite integrals, which will be given below. The table of indefinite integrals lists common functions, indicating the antiderivatives from which these functions are "made". As part of the tasks for finding the indefinite integral, such integrands are given that can be integrated directly without special efforts, that is, according to the table of indefinite integrals. In more complex problems, the integrand must first be transformed so that tabular integrals can be used.
Fact 2. 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 different constants from 1 to infinity, you need to write down a set of antiderivatives with an arbitrary constant C, 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 differentiating 4 or 3 or any other constant vanishes.
We set the integration problem: for a given function f(x) find such a function F(x), whose derivative is equal to f(x).
Example 1 Find the set of antiderivatives of a function
Decision. For this function, the antiderivative is the function
Function F(x) is called antiderivative for the function f(x) if the derivative F(x) is equal to f(x), or, which is the same thing, the differential F(x) is equal to f(x) dx, i.e.
(2)
Therefore, the function is antiderivative for the function . However, it is not the only antiderivative for . They are also functions
where With is an arbitrary constant. This can be verified by differentiation.
Thus, if there is one antiderivative for a function, then for it there is an infinite set of antiderivatives that differ by a constant summand. All antiderivatives for a function are written in the above form. This follows from the following theorem.
Theorem (formal statement of fact 2). If a F(x) is the antiderivative for the function f(x) on some interval X, then any other antiderivative for f(x) on the same interval can be represented as F(x) + C, where With is an arbitrary constant.
In the following example, we already turn to the table of integrals, which will be given in paragraph 3, after the properties of the indefinite integral. We do this before familiarizing ourselves with 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 when integrating.
Example 2 Find sets of antiderivatives:
Decision. 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, and we will study the table of indefinite integrals in full 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) at n= -1/4 find
Under the integral sign, they do not write the function itself f, and its product by the differential dx. This is done primarily to indicate which variable the antiderivative is being searched for. For example,
,
;
here in both cases the integrand is equal to , but its indefinite integrals in the considered cases turn out to be different. In the first case, this function is considered as a function of a 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.
The geometric meaning of the indefinite integral
Let it be required to find a curve y=F(x) and we already know that the tangent of the slope of the tangent at each of its points is given function f(x) abscissa of this point.
According to geometric sense derivative, tangent of the slope of the tangent at a given point on 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). Required function in the task F(x) is derived from f(x). The condition of the problem is 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 a F"(x)=f(x), then the graph of the function y=F(x) 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 is determined by an arbitrary constant (constant) of integration 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. The 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.
Solving integrals is an easy task, but only for the elite. This article is for those who want to learn to understand integrals, but know little or nothing about them. Integral... Why is it needed? How to calculate it? What are definite and indefinite integrals?
If the only use of the integral you know is to get something useful from hard-to-reach places with a hook in the shape of an integral icon, then welcome! Learn how to solve simple and other integrals and why you can't do without it in mathematics.
We study the concept « integral »
Integration was already known in Ancient Egypt. Of course not in modern form, but still. Since then, mathematicians have written a great many books on the subject. Particularly distinguished 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. mathematical analysis. Information about , which is also necessary for understanding integrals, is already in our blog.
Indefinite integral
Let's have some function f(x) .
The indefinite integral of the function f(x) such a function is called F(x) , whose derivative is equal to the function f(x) .
In other words, an integral is a reverse derivative or antiderivative. By the way, about how to read in our article.
The primitive exists for everyone 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 an integral is called integration.
Simple example:
In order not to constantly calculate the primitives elementary functions, it is convenient to summarize 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 calculate the area of \u200b\u200bthe figure, the mass of an inhomogeneous body passed through uneven movement path and more. It should be remembered that the integral is the sum of an infinitely large number of infinitely small terms.
As an example, imagine a graph of some function.
How to find the area of a figure limited schedule functions? With the help of an integral! Let's break the curvilinear trapezoid, bounded by the coordinate axes and the graph of the function, into infinitesimal segments. Thus, 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 the definite integral, which is written as follows:
The points a and b are called the 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 consider the properties of the indefinite integral, which will be useful in 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. Also true for the difference:
Properties of the Definite Integral
- Linearity:
- The sign of the integral changes if the limits of integration are reversed:
- At any points a, b and with:
We have already found out that the definite integral is the limit of the 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 consider the indefinite integral and examples with solutions. We offer you to independently understand the intricacies of the solution, and if something is not clear, ask questions in the comments.
To consolidate the material, watch a video on how integrals are solved in practice. Do not despair if the integral is not given immediately. Turn to a professional student service, and any triple or curvilinear integral over a closed surface will be within your power.
This article talks in detail about the main properties of a definite integral. They are proved using the concept of the Riemann and Darboux integral. The calculation of a definite integral passes, thanks to 5 properties. The rest of them are used to evaluate various expressions.
Before passing to the main properties of the definite integral, it is necessary to make sure that a does not exceed b .
Basic properties of a definite integral
Definition 1The function y \u003d f (x) , defined for x \u003d a, is similar to the fair equality ∫ a a f (x) d x \u003d 0.
Proof 1
From here we see that the value of the integral with coinciding limits is equal to zero. This is a consequence of the Riemann integral, because each integral sum σ for any partition on the interval [ a ; a ] and any choice of points ζ i equals zero, because x i - x i - 1 = 0 , i = 1 , 2 , . . . , n , so we get that the limit of integral functions is zero.
Definition 2
For a function integrable on the segment [ a ; b ] , the condition ∫ a b f (x) d x = - ∫ b a f (x) d x is satisfied.
Proof 2
In other words, if you change the upper and lower limits of integration in places, then the value of the integral will change the value to the opposite. This property is taken from the Riemann integral. However, the numbering of the division of the segment starts from the point x = b.
Definition 3
∫ a b f x ± g (x) d x = ∫ a b f (x) d x ± ∫ a b g (x) d x is used for integrable functions of the type y = f (x) and y = g (x) defined on the interval [ a ; b] .
Proof 3
Write the integral sum of the function y = f (x) ± g (x) for partitioning into segments with a given choice of points ζ i: σ = ∑ i = 1 n f ζ i ± g ζ i x i - x i - 1 = = ∑ i = 1 n f (ζ i) x i - x i - 1 ± ∑ i = 1 n g ζ i x i - x i - 1 = σ f ± σ g
where σ f and σ g are the integral sums of the functions y = f (x) and y = g (x) for splitting the segment. After passing to the limit at λ = m a x i = 1 , 2 , . . . , n (x i - x i - 1) → 0 we get that lim λ → 0 σ = lim λ → 0 σ f ± σ g = lim λ → 0 σ g ± lim λ → 0 σ g .
From Riemann's definition, this expression is equivalent.
Definition 4
Taking the constant factor out of the sign of a definite integral. An integrable function from the interval [ a ; b ] with an arbitrary value of k has a valid inequality of the form ∫ a b k · f (x) d x = k · ∫ a b f (x) d x .
Proof 4
The proof of the property of a definite integral is similar to the previous one:
σ = ∑ i = 1 n k f ζ i (x i - x i - 1) = = k ∑ i = 1 n f ζ i (x i - x i - 1) = k σ f ⇒ lim λ → 0 σ = lim λ → 0 (k σ f) = k lim λ → 0 σ f ⇒ ∫ a b k f (x) d x = k ∫ a b f (x) d x
Definition 5
If a function of the form y = f (x) is integrable on an interval x with a ∈ x , b ∈ x , we obtain ∫ a b f (x) d x = ∫ a c f (x) d x + ∫ c b f (x) d x .
Proof 5
The property is considered to be valid for c ∈ a ; b , for c ≤ a and c ≥ b . The proof is carried out similarly to the previous properties.
Definition 6
When a function has the ability to be integrable from the segment [ a ; b ] , then this is feasible for any internal segment c ; d ∈ a; b.
Proof 6
The proof is based on the Darboux property: if points are added to an existing partition of a segment, then the lower Darboux sum will not decrease, and the upper one will not increase.
Definition 7
When a function is integrable on [ a ; b ] from f (x) ≥ 0 f (x) ≤ 0 for any value of x ∈ a ; b , then we get that ∫ a b f (x) d x ≥ 0 ∫ a b f (x) ≤ 0 .
The property can be proved using the definition of the Riemann integral: any integral sum for any choice of partition points of the segment and points ζ i with the condition that f (x) ≥ 0 f (x) ≤ 0 is non-negative.
Proof 7
If the functions y = f (x) and y = g (x) are integrable on the segment [ a ; b ] , then the following inequalities are considered valid:
∫ a b f (x) d x ≤ ∫ a b g (x) d x , f (x) ≤ g (x) ∀ x ∈ a ; b ∫ a b f (x) d x ≥ ∫ a b g (x) d x , f (x) ≥ g (x) ∀ x ∈ a ; b
Thanks to the assertion, we know that the integration is admissible. This corollary will be used in the proof of other properties.
Definition 8
For an integrable function y = f (x) from the segment [ a ; b ] we have a valid inequality of the form ∫ a b f (x) d x ≤ ∫ a b f (x) d x .
Proof 8
We have that - f (x) ≤ f (x) ≤ f (x) . From the previous property, we obtained that the inequality can be integrated term by term and it corresponds to an inequality of the form - ∫ a b f (x) d x ≤ ∫ a b f (x) d x ≤ ∫ a b f (x) d x . This double inequality can be written in another form: ∫ a b f (x) d x ≤ ∫ a b f (x) d x .
Definition 9
When the functions y = f (x) and y = g (x) are integrated from the segment [ a ; b ] for g (x) ≥ 0 for any x ∈ a ; b , we obtain an inequality of the form m ∫ a b g (x) d x ≤ ∫ a b f (x) g (x) d x ≤ M ∫ a b g (x) d x , where m = m i n x ∈ a ; b f (x) and M = m a x x ∈ a ; b f (x) .
Proof 9
The proof is done in a similar way. M and m are considered to be the largest and the smallest value function y = f (x) , defined from the segment [ a ; b ] , then m ≤ f (x) ≤ M . It is necessary to multiply the double inequality by the function y = g (x) , which will give the value of the double inequality of the form m g (x) ≤ f (x) g (x) ≤ M g (x) . It is necessary to integrate it on the segment [ a ; b ] , then we obtain the assertion to be proved.
Consequence: For g (x) = 1, the inequality becomes m b - a ≤ ∫ a b f (x) d x ≤ M (b - a) .
First average formula
Definition 10For y = f (x) integrable on the interval [ a ; b ] with m = m i n x ∈ a ; b f (x) and M = m a x x ∈ a ; b f (x) there is a number μ ∈ m ; M , which fits ∫ a b f (x) d x = μ · b - a .
Consequence: When the function y = f (x) is continuous from the segment [ a ; b ] , then there is such a number c ∈ a ; b , which satisfies the equality ∫ a b f (x) d x = f (c) b - a .
The first formula of the average value in a generalized form
Definition 11When the functions y = f (x) and y = g (x) are integrable from the segment [ a ; b ] with m = m i n x ∈ a ; b f (x) and M = m a x x ∈ a ; b f (x) , and g (x) > 0 for any value of x ∈ a ; b. Hence we have that there is a number μ ∈ m ; M , which satisfies the equality ∫ a b f (x) g (x) d x = μ · ∫ a b g (x) d x .
Second mean value formula
Definition 12When the function y = f (x) is integrable from the segment [ a ; b ] , and y = g (x) is monotonic, then there is a number that c ∈ a ; b , where we obtain a fair equality of the form ∫ a b f (x) g (x) d x = g (a) ∫ a c f (x) d x + g (b) ∫ c b f (x) d x
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