The indefinite integral of the derivative of the integrand is equal to. Integrals for Dummies: How to Solve, Calculation Rules, Explanation. The indefinite integral of the differential of some function is equal to the sum of this function and an arbitrary constant

Antiderivative and indefinite integral.

The antiderivative of a function f (x) on the interval (a; b) is a function F (x) such that equality holds for any x from a given interval.

If we take into account the fact that the derivative of the constant С 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 a 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

Simplest properties of integrals

1. The derivative of the result of integration 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 outside the sign of the indefinite integral.

4. The indefinite integral of the sum / difference of functions is equal to the sum / difference of the indefinite integrals of the functions.

Intermediate equalities of the first and second properties of the indefinite integral are given for clarification.

To prove the third and fourth properties, it is enough 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 of the differentiation problem, and there is a very close connection between these problems:

the first property allows one to check the integration. To check the correctness of the performed integration, it is enough to calculate the derivative of the obtained result. If the function obtained as a result of differentiation turns out to be equal to the integrand, then this will mean that the integration was carried out correctly;

the second property of the indefinite integral allows us to find its antiderivative from the known differential of the function. Direct calculation of indefinite integrals is based on this property.

1.4 Invariance of forms of integration.

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).

of the function f (x) is reduced to calculating the integral of the differential form f.w, where

An explicit f-la for r (x) is given below. The matching condition is .

here Tg means the operator of shift on X by means of gОG: Tgf (x) = f (g-1x). Let X = G be a topology, a group acting on itself by left translations. I. and. exists if and only if G is locally compact (in particular, on infinite-dimensional groups I. and. does not exist). For a subset of I. and. characteristic function cA (equal to 1 on A and 0 outside A) defines the left Xaara 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 determined up to a 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 (a mapping that preserves the group property) DG of a group G into a group (with respect to multiplication) puts. numbers for which

where dmr and dmi are 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 coincide. 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 an n-form. In local coordinates to compute

forms qi, any matrix realization of the group G can be used: the matrix 1-form g-1dg is left-invariant, and its coefficient. are left-invariant scalar 1-forms, from which the required basis is selected. For example, the full matrix group GL (n, R) is unimodular and the Haar measure on it is given by the form. Let be X = G / H is a homogeneous space for which a locally compact group G is a transformation group, and a closed subgroup H is a stabilizer of some point. In order for an I. and. To exist on X, it is necessary and sufficient that for all hОH the equality DG (h) = DH (h) holds. In particular, this is true in the case when H is compact or semisimple. Complete theory I. and. does not exist on infinite-dimensional manifolds.

Change of variables.

This article details the basic properties of a definite integral. They are proved using the concept of the Riemann and Darboux integral. The calculation of the definite integral takes place thanks to 5 properties. The rest of them are used to evaluate various expressions.

Before proceeding to the basic properties of a definite integral, it is necessary to make sure that a does not exceed b.

Basic properties of a definite integral

Definition 1

The function y = f (x), defined at x = a, is similar to the valid equality ∫ a a f (x) d x = 0.

Proof 1

Hence 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 is equal to 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 that is 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 the upper and lower limits of integration are changed in places, then the value of the integral will change its value to the opposite. This property is taken from the Riemann integral. However, the numbering of the division of the segment comes 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 down the integral sum of the function y = f (x) ± g (x) for partitioning into segments with the given choice of points ζ i: σ = ∑ i = 1 nf ζ i ± g ζ i xi - xi - 1 = = ∑ i = 1 nf (ζ i) xi - xi - 1 ± ∑ i = 1 ng ζ i xi - xi - 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 the partition of the segment. After passing to the limit at λ = m a x i = 1, 2,. ... ... , n (x i - x i - 1) → 0 we obtain 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

Carrying out a constant factor beyond 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 the definite integral is similar to the previous one:

σ = ∑ i = 1 nk f ζ i (xi - xi - 1) = = k ∑ i = 1 nf ζ i (xi - xi - 1) = k σ f ⇒ lim λ → 0 σ = lim λ → 0 (k σ f) = k lim λ → 0 σ f ⇒ ∫ abk f (x) dx = k ∫ abf (x) dx

Definition 5

If a function of the form y = f (x) is integrable on an interval x with a ∈ x, b ∈ x, we obtain that ∫ 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 true for c ∈ a; b, for c ≤ a and c ≥ b. The proof is similar to the previous properties.

Definition 6

When the function has the ability to be integrable from the segment [a; b], then it is doable for any inner segment c; d ∈ a; b.

Proof 6

The proof is based on the Darboux property: if we add points to the existing partition of a segment, then the lower Darboux sum will not decrease, and the upper one will not increase.

Definition 7

When the function is integrable on [a; b] from f (x) ≥ 0 f (x) ≤ 0 for any value of x ∈ a; b, then we obtain 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, we obtain non-negative.

Proof 7

If the functions y = f (x) and y = g (x) are integrable on the interval [a; b], then the following inequalities are considered to be true:

∫ a b f (x) d x ≤ ∫ a b g (x) d x, if and f (x) ≤ g (x) ∀ x ∈ a; b ∫ a b f (x) d x ≥ ∫ a b g (x) d x, if and f (x) ≥ g (x) ∀ x ∈ a; b

Thanks to the statement, we know that integration is admissible. This corollary will be used to prove other properties.

Definition 8

With 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 carried out in a similar way. M and m are considered the largest and the smallest value of the function y = f (x) determined 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.

Corollary: For g (x) = 1, the inequality takes the form m b - a ≤ ∫ a b f (x) d x ≤ M (b - a).

First mean value formula

Definition 10

For y = f (x), integrable on the segment [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.

Corollary: 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.

First mean value formula in generalized form

Definition 11

When 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 12

When the function y = f (x) is integrable from the segment [a; b], and y = g (x) is monotone, then there is a number that c ∈ a; b, where we obtain a valid 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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These properties are used to carry out transformations of the integral with the aim of reducing 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 some 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. The property is a combination of properties 4 and 5:

Moreover, a ≠ 0 ˄ b ≠ 0

7. Property of invariance 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 consider an example:

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

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

The main task of differential calculus is finding the derivative f '(x) or differential df =f '(x)dx function f (x). In integral calculus, the inverse problem is solved. For a given function f (x) it is required 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) with respect to the known derivative (differential) of this function. Integral calculus has numerous applications in geometry, mechanics, physics, and engineering. It provides a general method for finding areas, volumes, centers of gravity, etc.

Definition. FunctionF (x), is called the antiderivative for 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 on the segment [a;b] functionf (x) has the 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. The aggregateF (x) +C of all antiderivatives of the functionf (x) on the set X is called an indefinite integral and is denoted by:

- (1)

In formula (1) f (x)dx called the integrand,f (x) is the integrand, x is the variable of integration, a С - constant of integration.

Consider the properties of the indefinite integral following 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 some function is equal to the sum of this function and an arbitrary constant:

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

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

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

6 (invariance of the integration formulas). Any integration formula retains its form if the integration variable is replaced by any differentiable function of this variable:

whereu is a differentiable function.

Indefinite integrals table.

Let us give basic rules for integrating functions.

Let us give table of basic indefinite integrals.(Note that here, as in differential calculus, the letter u can denote 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 of the table of integrals, which have no analogue in the table of derivatives, are checked by differentiating their right-hand sides.

Variable change and integration by parts in the indefinite integral.

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

which is not tabular. The essence of the substitution method is that in the integral the variable NS replace with 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 some set T and let X be the set of values ​​of this function, on which the functionf (x). Then if on the set X the functionf (

In differential calculus, the following problem is solved: under the given function ƒ (x) find its derivative(or differential). The integral calculus solves the inverse problem: find the function F (x), knowing its derivative F "(x) = ƒ (x) (or differential). The required 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

where C is a constant, since

Theorem 29. 1. If the function F (x) is the 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) + С, where С is a constant number.

▲ The function F (x) + C is the antiderivative of ƒ (x).

Indeed, (F (x) + C) "= F" (x) = ƒ (x).

Let Ф (x) be some other, different from F (x), antiderivative of the function (x), that is, Φ "(x) = ƒ (x). Then for any x є (a; b) we have

This means (see Corollary 25.1) that

where C is a constant number. Therefore, Ф (х) = F (x) + С. ▼

The set of all first-inverse functions F (x) + C for ƒ (x) is called the 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 - the integrand, NS - variable of integration, ∫ -indefinite integral sign.

The operation of finding the indefinite integral of a function is called the integration of this function.

A geometrically indefinite integral is a family of "parallel" curves y = F (x) + C (each numerical value of C corresponds to a certain family curve) (see Fig. 166). The graph of each antiderivative (curve) is called integral curve.

Does an indefinite integral exist for every function?

There is a theorem asserting 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).

Due to this property, the correctness of integration is checked by differentiation. For example, equality

∫ (3x 2 + 4) dx = x h + 4x + C

true, since (x 3 + 4x + C) "= 3x 2 + 4.

2. The indefinite integral of the differential of some 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 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 (х) is an arbitrary function with a continuous derivative.

▲ Let x be an independent variable, ƒ (x) a continuous function, and F (x) its first image. Then

We now put u = φ (x), where φ (x) is a continuously differentiable function. Consider a complex function F (u) = F (φ (x)). Since the form of the first differential of the function is invariant (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 with a continuous derivative.

So, from the formula by replacing x with u (u = φ (x)), we obtain

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. Basic Indefinite Integral Table

Taking advantage of the fact that integration is the inverse action of differentiation, it is possible to obtain a table of basic integrals by inverting the corresponding formulas of the 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 main methods of integration.

Integrals in the table below are called tabular. You should know them 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 the primitive (i.e., integrating a function) are reduced to specifying techniques that bring a given (sought) integral to a tabular one. Therefore, it is necessary to know tabular integrals and be able to recognize them.

Note that in the table of basic integrals the variable of integration and can denote both the independent variable and the function of the independent variable (in accordance with 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). НоMeans

So Formula 2 is correct. Similarly, let's check Formula 15:

Basic integral table

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