An indefinite integral from the derivative of the integrand function is equal. 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 arbitrary constant

Pred-like and indefinite integral.

The primitive function f (x) on the interval (A; b) is called such a function f (x), which is performed for any x from the specified gap.

If you take into account the fact that the derivative of the constant C is zero, then equality is right . Thus, the function f (x) has many primitive f (x) + c, for an arbitrary constant C, and these first-shaped differ from each other into an arbitrary constant value.

All many primary functions f (x) are called an uncertain integral of this function and is indicated .

The expression is called a concintive expression, and F (x) - the integrand function. The integrand is the differential function f (x).

The action of finding an unknown function according to its given differential is called uncertain integration, because the result of integration is not one function f (x), but the set of its primitive F (x) + c.

Table integrals

The simplest properties of the integrals

1. The derivative of the integration result is equal to the integrated function.

2. An indefinite integral of the differential function is equal to the sum of the function itself and an arbitrary constant.

3. The coefficient can be made for a sign of an undefined integral.

4. An indefinite integral of the amount / difference between functions is equal to the amount / difference of uncertain integrals of functions.

Interim equals of the first and second properties of an uncertain integral are given to explanation.

To prove the third and fourth properties, it is sufficient to find derivatives from the right parts of equality:

These derivatives are equal to the inhibitory functions, which is proof by virtue of the first property. It is used in the last transitions.

Thus, the integration task is the inverse differentiation problem, and there is very close relationship between these tasks:

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

the second property of an indefinite integral allows you to find its primitive function on a well-known differential. On this property, the direct calculation of uncertain integrals is based.

1.4. Invariance of integration forms.

Invariant integration - the type of integration for functions, the argument of which are the elements of a group or the point of a homogeneous space (any point of such space can be translated into another specified action of the group).

functions f (x) comes down to calculating the integral from the differential form F.W, where

The explicit F-la for R (x) is given below. The condition of coordination is .

here TG is a shift operator to X using GOG: TGF (X) \u003d F (G-1X). Let x \u003d g be a topology, a group acting on itself left shifts. I. and. It exists if and only if G is locally compact (in particular, on infinite-dimensional groups I. and. does not exist). For subset I. and. The characteristic functions of the Ca (equal to 1 per a and 0 outside A) sets the left measure of XAAR M (A). The defining property of this measure is its invariance with left shifts: M (G-1a) \u003d M (A) for all GOG. The left measure of Haar on the group is definitely defined with an accuracy to put, a scalar multiplier. If the Mera Haar M is known, then I. and. Functions f give formula . Similar properties have the right measure Haar. There is a continuous homomorphism (the display that stores the group property) DG group G to the group (relative to multiplication) will be posted. numbers for which

where DMR and DMI are the right and left hairdu measures. DG (G) Named function. Module group G. If, then group G Naz. unimodular; In this case, the right and left hairdi measures coincide. Compact, semisimple and nilpotent (in particular, commutative) groups are unimulated. If G is a N-dimensional group of Lee and Q1, ..., qn - the basis in the space of left-ring 1-forms per G, then the left measure of Haar on G is set by the N-form. In local coordinates for calculating

forms Qi can be used by any matrix realization of the group G: the matrix 1-form G-1DG left-invariant, and its coeffic. There are left-invariant scalar 1-forms, of which the desired basis is selected. For example, the full matrix group GL (N, R) is unimuscable and the Haara measure is set to form. Let be X \u003d g / h is a homogeneous space for which the locally compact group G is a group of transformations, and a closed subgroup H is a stabilizer of some point. In order to exist, I. and., It is necessary and enough to ensure that the equality Dg (H) \u003d DH (H) is performed for all. In particular, this is true in the case when n compact or semisimple. Full theory I. and. There are no infinite-dimensional manifolds.

Replacing variables.

This article tells in detail the basic properties of a particular integral. They are proved by the concept of the integral of Riemann and Darbu. Calculation of a specific integral passes, thanks to 5 properties. The remaining of them are used to estimate various expressions.

Before moving to the main properties of a specific integral, it is necessary to make sure that A does not exceed b.

The main properties of a specific integral

Definition 1.

The function y \u003d f (x), defined at x \u003d A, similar to the equitable equality ∫ a a f (x) d x \u003d 0.

Proof 1.

From here we see that the integral value with the coinciding limits is zero. This is a consequence of the Riemann integral, because each integral sum Σ for any partition at the interval [A; A] and any selection of points ζ i is zero, because x i - x i - 1 \u003d 0, i \u003d 1, 2 ,. . . , n, it means that we obtain that the limit of integral functions is zero.

Definition 2.

For a function integrated on the segment [a; b], the condition ∫ a b f (x) d x \u003d - ∫ b a f (x) d x is satisfied.

Proof 2.

In other words, if you change the upper and lower limit of integration in places, the integral value will change the value to the opposite. This property is taken from the integral of Riemann. However, the numbering of the splitting of the segment comes from the point x \u003d b.

Definition 3.

∫ a b f x ± g (x) d x \u003d ∫ a b f (x) d x ± ∫ a b g (x) d x is used for integrable functions like y \u003d f (x) and y \u003d g (x) defined on the segment [a; b].

Proof 3.

Record the integrated sum of the function y \u003d f (x) ± g (x) to split into segments with a given choice of points ζ i: σ \u003d Σ i \u003d 1 nf ζ i ± g ζ i · xi - xi - 1 \u003d \u003d σ i \u003d 1 nf (ζ i) · xi - xi - 1 ± σ i \u003d 1 ng ζ i · xi - xi - 1 \u003d σ f ± σ g

where σ f and σ g are the integral sums of the functions y \u003d f (x) and y \u003d g (x) to split the segment. After transition to the limit at λ \u003d m a x i \u003d 1, 2 ,. . . , N (x i - x i - 1) → 0 We obtain that Lim Λ → 0 Σ \u003d Lim Λ → 0 Σ F ± Σ G \u003d Lim Λ → 0 Σ g ± Lim Λ → 0 Σ g.

From the definition of Riemann, this expression is equivalent.

Definition 4.

Reaching a constant factor for a sign of a certain integral. Integrable function from the interval [a; b] with an arbitrary value K has a fair inequality of the form ∫ a b k · f (x) d x \u003d k · ∫ a b f (x) d x.

Proof 4.

Proof of the properties of a specific integral similarly to the previous one:

σ \u003d σ i \u003d 1 nk · f ζ i · (xi - xi - 1) \u003d k · σ i \u003d 1 nf ζ i · (xi - xi - 1) \u003d k · σ f ⇒ lim λ → 0 σ \u003d Lim Λ → 0 (k · Σ f) \u003d k · lim λ → 0 Σ F ⇒ ∫ ABK · F (X) DX \u003d k · ∫ ABF (X) DX

Definition 5.

If the function of the form y \u003d f (x) is integrated on the interval x with a ∈ x, b ∈ X, we obtain that ∫ a b f (x) d x \u003d ∫ a c f (x) d x + ∫ c b f (x) d x.

Proof 5.

The property is considered valid for c ∈ A; b, for c ≤ a and c ≥ b. Proof is carried out similarly to previous properties.

Definition 6.

When the function has the ability to be integrated from the segment [a; b], then it is done for any inner segment C; d ∈ A; b.

Proof 6.

The proof is based on the Darbé property: if the existing splitting of the segment is added to add points, then the lower amount of Darboux will not decrease, and the upper will not increase.

Definition 7.

When the function is integrated to [a; b] from f (x) ≥ 0 f (x) ≤ 0 at any meaning x ∈ A; b, then we obtain that ∫ a b f (x) d x ≥ 0 ∫ A b f (x) ≤ 0.

The property can be proven using the determination of the Riemann integral: any integral amount for any selection of the separation points of the segment and points ζ i with the condition that f (x) ≥ 0 f (x) ≤ 0, we obtain nonnegative.

Proof 7.

If the functions y \u003d f (x) and y \u003d g (x) are integrable on the segment [a; b], then the following inequalities are considered fair:

∫ a b f (x) d x ≤ ∫ a b g (x) d x, e s l and f (x) ≤ g (x) ∀ x ∈ A; b ∫ a b f (x) d x ≥ ∫ a b g (x) d x, e s l and f (x) ≥ g (x) ∀ x ∈ A; B.

Thanks to the statement, we know that integration is permissible. This investigation will be used in the proof of other properties.

Definition 8.

With the integrable function y \u003d f (x) from the segment [a; b] We have a fair 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, it was obtained that inequality can be integrated and corresponds to it 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 recorded in the other form: ∫ a b f (x) d x ≤ ∫ a b f (x) d x.

Definition 9.

When the functions y \u003d f (x) and y \u003d g (x) are integrated from the segment [a; b] for g (x) ≥ 0 at any x ∈ A; b, we obtain the 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 \u003d m i n x ∈ A; b f (x) and m \u003d m a x x ∈ A; B f (x).

Proof 9.

Similarly proof is proof. M and M are considered the greatest and smallest value of the function y \u003d f (x), determined from the segment [A; b], then m ≤ f (x) ≤ m. It is necessary to multiply the double inequality on the function y \u003d 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 will get a proven statement.

Consequence: At G (x) \u003d 1, the inequality takes the form M · b - a ≤ ∫ a b f (x) d x ≤ m · (b - a).

The first middle formula

Definition 10.

At y \u003d f (x) integrated on the segment [a; b] with m \u003d m i n x ∈ A; b f (x) and m \u003d m a x x ∈ A; b f (x) there is a number μ ∈ M; M, which is suitable for ∫ a b f (x) d x \u003d μ · b - a.

Consequence: When the function y \u003d 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 \u003d f (c) · b - a.

The first middle formula in generalized formula

Definition 11.

When the functions y \u003d f (x) and y \u003d g (x) are integrable from the segment [A; b] with m \u003d m i n x ∈ A; b f (x) and m \u003d m a x x ∈ A; b f (x), and g (x)\u003e 0 for any meaning x ∈ A; b. From here we have that there is a number μ ∈ M; M, which satisfies the equality ∫ a b f (x) · g (x) d x \u003d μ · ∫ a b g (x) d x.

Second medium formula

Definition 12.

When the function y \u003d f (x) is an integrable of a segment [a; b], and y \u003d g (x) is monotonous, 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 \u003d 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 implement the integral transformations in order to bring it to one of the elementary integrals and further computation.

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

2. The differential of an indefinite integral is equal to the initial expression:

3. An indefinite integral from the differential of some function is equal to the sum of this function and arbitrary constant:

4. A permanent multiplier can be made for an integral sign:

And A ≠ 0

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

6. The property is a combination of properties 4 and 5:

And A ≠ 0 B ≠ 0

7. Property of invariance of an indefinite integral:

If, then

8. Property:

If, then

In fact, this property is a special integration case using a variable replacement method, which is described in more detail in the next section.

Consider an example:

At first we applied property 5, then property 4, then used the table of primitive and obtained the result.

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

The main task of differential calculus is the finding of the derivative f '(x) or Differential df \u003d.f '(x)dXfunctions f (x). In the integral calculation is solved inverse. According to a given function f (x.) It is required to find such a function. F (x) what F '(x) \u003df (x) or dF (x) \u003d.F '(x)dX \u003d.f (x)dX.

In this way, the main task of integral calculus Is restoring a function F (x) According to the known derivative (differential) of this function. Integrative calculation has numerous applications in geometry, mechanics, physics and technology. It gives a general method of finding space, volumes, centers of gravity, etc.

Definition. FunctionF (x), called primitive for functionf (x) on the set x, if it is differentiable for any andF '(x) \u003d.f (x) ordF (x) \u003d.f (x)dX.

Theorem. Any continuous on the segment [a;b] Functionf (x) has a primitive in this segmentF (x).

Theorem. If aF 1 (x) I.F 2 (x) - two different primary one and the same functionf (x) on the set x, then they differ from each other constant terms, i.e.F 2 (x) \u003d.F 1.x) +.C, where C is constant.

Uncertain integral, its properties.

Definition. TotalF (x) +.With all the primitive functionsf (x) on the set X is called an uncertain integral and designated:

- (1)

In formula (1) f (x)dXcalled a clear expressionf (x) - integrated function, x - integration variable,but C - constant integration.

Consider the properties of an uncertain integral arising from its definition.

1. A derivative of an indefinite integral is equal to the integrand function, the differential of an indefinite integral is equal to the integrative expression:

and.

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

3. A permanent multiplier A (A ≠ 0) can be made for a sign of an undefined integral:

4. The indefinite integral from the algebraic amount of the final number of functions is equal to the algebraic amount of integrals from these functions:

5. If aF (x) - a primitive functionf (x), then:

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

whereu is a differentiable function.

Table of uncertain integrals.

Here basic rules for integrating functions.

Here table of basic uncertain integrals. (We note that here, as in the differential calculus, the letter u. may indicate as an independent variable (u \u003d.x)and the function from an independent variable (u \u003d.u (x)).)

(n ≠ -1). (A\u003e 0, A ≠ 1). (A ≠ 0). (A ≠ 0). (| U |\u003e | A |). (| U |< |a|).

Integrals 1 - 17 called tables.

Some of the above formulas of the integral table that have no analogue in the derivative table are checked by the differentiation of their right-hand parts.

Replacing variable and integration in parts in an indefinite integral.

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

which is not tabular. The essence of the method of substitution is that the integral variable h. Replace the variable t. according to the formula x \u003d φ (t) From dx \u003d φ '(t)dT.

Theorem. Let the functionx \u003d φ (t) defined and differentiable at some set T and let x - the set of values \u200b\u200bof this function, on which the function is definedf (x). Then if on the set x functionf (

The task is solved in differential calculation: under Anna Functions ƒ (x) Find its derivative(or differential). Integrated calculus solves the inverse problem: find the function f (x), knowing its derivative F "(x) \u003d ƒ (x) (or differential). The desired function f (x) is called the primitive function ƒ (x).

Function F (x) is called predo-shapedfunctions ƒ (x) on the interval (A; b), if for any x є (a; b) equality is performed

F "(x) \u003d ƒ (x) (or df (x) \u003d ƒ (x) dx).

for example, the primitive function y \u003d x 2, x є r, is a function, since

Obviously, any functions will also be primitive

where C is constant because

TEPEMA 29. 1. If the function f (x) is a primitive function ƒ (x) to (a; b), then the set of all very primitive for ƒ (x) is given by the formula F (x) + C, where C is a constant number.

▲ Function F (x) + C is a primitive ƒ (x).

Indeed, (f (x) + c) "\u003d f" (x) \u003d ƒ (x).

Let F (x) be some other different from f (x), the primitive function ƒ (x), i.e. f "(x) \u003d ƒ (x). Then for any x є (a; b) we have

And this means (see Corollary 25. 1) that

where C is a constant number. Consequently, f (x) \u003d f (x) + C. ▼

The set of all the parameters of the functions f (x) + C for ƒ (x) called uncertain integral from function ƒ (x)and denotes symbol∫ ƒ (x) DX.

Thus, by definition

∫ ƒ (x) dx \u003d f (x) + c.

Here ƒ (x) called disgrave function, ƒ (x) DX - a clear expressionx - variable integration, ∫ -sign an indefinite integral.

The operation of finding an indefinite integral from a function is called integrating this function.

The geometrically indefinite integral is a family of "parallel" curves y \u003d f (x) + C (a certain curve of the family corresponds to each numerical value of C corresponds (see Fig. 166). The graph of each primitive (curve) is called integral curve.

Is there an indefinite integral for any function?

There is a theorem that claims that "any continuous on (a; b) function has a primitive effect on this gap", and therefore, and an indefinite integral.

We note a number of properties of an indefinite integral arising from its definition.

1. The differential from an undefined integral is equal to the initial expression, and the derivative of an uncertain integral is equal to the integrand function:

d (∫ ƒ (x) dx) \u003d ƒ (x) dh, (∫ ƒ (x) dx) "\u003d ƒ (x).

Deisitive, D (∫ ƒ (x) dx) \u003d d (f (x) + s) \u003d df (x) + d (c) \u003d f "(x) dx \u003d ƒ (x) dx

(∫ ƒ (x) dx) "\u003d (f (x) + c)" \u003d f "(x) +0 \u003d ƒ (x).

Blugging this property is the correct integration is verified by differentiation. For example, equality

∫ (3x 2 + 4) dx \u003d x z + 4x + with

right, since (x 3 + 4x + s) "\u003d 3x 2 +4.

2. The first integral of the diffpectivity of some function is equal to the sum of this function and arbitrary constant:

∫df (x) \u003d f (x) + c.

Really,

3. A permanent multiplier can be made for the integral sign:

α ≠ 0 - constant.

Really,

(Put with 1 / a \u003d p.)

4. An indefinite integral from the angeboqueous amount of the finite number of continuous functions is equal to the surgefulness of the integral sum of the terms of the functions:

Let F "(x) \u003d ƒ (x) and g" (x) \u003d g (x). Then

where C 1 ± C 2 \u003d s.

5. (invariance of the integration formula).

If a where u \u003d φ (x) is an arbitrary function having a continuous derivative.

▲ Let X be an independent variable, ƒ (x) - a continuous function and f (x) - its peak. Then

Now we can register U \u003d F (x), where F (x) is a continuously differentiable function. Consider the complex function f (u) \u003d f (φ (x)). Due to the invaerability of the form of the first differential function (see p. 160) we have

From here ▼

Thus, the formula for an indefinite integral remains fair regardless of whether the integration variable is an independent variable or any function from it having a continuous derivative.

So, from the formula by replacing x on u (u \u003d φ (x)) we get

In particular,

Example 29.1. Find integral

where C \u003d C1 + C 2 + C 3 + C 4.

Example 29.2. Find an integral solution:

• 29.3. Table of basic indefinite integrals

Using the fact that the integration is an effect, inverse differentiation, you can get a table of basic integrals by referring to the corresponding formulas of the diffpectivity of calculus (differential table) and using the properties of an indefinite integral.

for example, as

d (sin u) \u003d cos u. du,

The conclusion of a number of table formulas will be given when considering the main integration methods.

Integrals in the table below are called tabular. They should be known by heart. In the integral calculus there are no simple and universal rules for finding the primary from elementary functions, as in differential calculus. Methods for finding peppercase (i.e. function integration) are reduced to the instructions of the receptions leading this (desired) integral to the table. Consequently, you need to know the table integrals and be able to recognize them.

Note that in the table of the main integrals, the integration variable can be denoted as an independent variable and the function from an independent variable (the property of the invariance of the integrate formula).

In the justice of the formulas below, it is possible to make sure that the opposite of the right-hand side, which will be equal to the initial expression on the left side of the formula.

We prove, for example, the justice of formula 2. The function 1 / U is defined and continuous for all values \u200b\u200band other than zero.

If u\u003e 0, then ln | u | \u003d lnu, then therefore

Esley U.<0, то ln|u|=ln(-u). Но So

So, formula 2 is true. Angurtherous, Consider Formula 15:

Table Output Integrals

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