Lesson on the topic; The antiderivative of a function. Indefinite integral. Antiderivative and indefinite integral, properties Antiderivative of function and indefinite integral

home » new time Lesson on the topic; The antiderivative of a function. Indefinite integral. Antiderivative and indefinite integral, properties Antiderivative of function and indefinite integral

The main task differential calculus is to find the differential of a given function or its derivative. The integral calculus solves the inverse problem: according to a given differential, and, consequently, the derivative of an unknown function F(x), you need to define this function. In other words, having the expression

or respectively

Where f(x) is a known function, you need to find the function F(x). Required function F(x) it is called antiderivative function with respect to function f(x). For simplicity, we will assume that equality (1) holds on some finite or infinite interval.

Definition: An antiderivative function for a given function f(x) on this interval is called such a function F(x), whose derivative is f(x) or whose differential is f(x)dx within the considered interval.

For example, one of the antiderivatives for a function will be , because . The antiderivative function is not unique, since etc., and therefore the functions and so on. are also antiderivatives for the function . Therefore, this function has an infinite number of antiderivatives.

In our example, every two antiderivatives differed from each other by some constant term. Let us show that this will also take place in the general case.

Theorem: Two different antiderivatives of the same function defined on a certain interval differ from each other on this interval by a constant term.

Proof: Indeed, let f(x) is some function defined on the interval , And F 1 (x), F 2 (x) are its primitives, i.e.

And .

From here .

y=F 1 (x)

y=F 2 (x)

F 1 (x)

F 2 (x)

WITH

M 2

M 1

Rice. 1.

But if two functions have the same derivatives, then these functions differ from each other by a constant term. Hence,

F 1 (x) - F 2 (x) \u003d C,

Where WITH is a constant value. The theorem has been proven.

Consider a geometric illustration. If y \u003d F 1 (x) and Y \u003d F 2 (x)

Antiderivatives of the same function f(x), then the tangents to their graphs at points with a common abscissa X parallel to each other (Fig. 1):

tga = = f(x).

In this case, the distance between these curves along the axis OU remains constant: F 2 (x) - F 1 (x) \u003d C, those. these curves are in some sense "parallel" to each other.

Consequence: Adding to some antiderivative function f(x) defined on the interval , all possible constants WITH, we get all antiderivatives for the function f(x).

Indeed, if F(x) there is an antiderivative function for f(x), then the function F(x)+C, Where WITH- any constant will also be an antiderivative function f(x), because .

On the other hand, we have proved that every antiderivative of a function f(x) can be obtained from the function F(x) by adding to it a properly chosen constant term WITH.

Therefore, the expression F(x) + C, Where , (2)

Where F(x)- some antiderivative for the function f(x), exhausts the entire collection of antiderivatives for a given function f(x).

In what follows, we will assume, unless explicitly stated otherwise, that the function under consideration f(x) is defined and continuous on some finite or infinite interval .

Let us now introduce the basic concept of the integral calculus - the concept of an indefinite integral.

Definition: General expression for all antiderivatives of a given continuous function f(x) is called the indefinite integral of the function f(x) or from the differential expression f(x)dx and is denoted by the symbol .

At the same time, the function f(x) is called the integrand, and the expression f(x)dx is called the integrand.

According to the definition of the indefinite integral, one can write

, (3)

From 4

From 3

From 2

From 1

Rice. 2.

Where

, constant WITH can take on any value and is therefore called an arbitrary constant.

Example. As we have seen, for a function one of the antiderivatives is the function . That's why .

Geometrically indefinite integral y=F(x)+C is a family of "parallel" curves (Fig. 2).

A review of methods for calculating indefinite integrals is presented. The main methods of integration are considered, which include integration of the sum and difference, taking the constant out of the integral sign, changing the variable, and integrating by parts. Also considered are special methods and techniques for integrating fractions, roots, trigonometric and exponential functions.

Content

Sum (difference) integration rule

Taking the constant out of the integral sign

Let c be a constant independent of x. Then it can be taken out of the integral sign:

Variable substitution

Let x be a function of a variable t , x = φ(t) , then
.
Or vice versa, t = φ(x) ,
.

With the help of a change of variable, you can not only calculate simple integrals, but also simplify the calculation of more complex ones.

Rule of Integration by Parts

Integration of fractions (rational functions)

Let's introduce a notation. Let P k (x), Q m (x), R n (x) denote polynomials of degrees k, m, n , respectively, with respect to the variable x .

Consider an integral consisting of a fraction of polynomials (the so-called rational function):

If k ≥ n, then first you need to select the integer part of the fraction:
.
The integral of the polynomial S k-n (x) is calculated from the table of integrals.

The integral remains:
, where m< n .
To calculate it, the integrand must be decomposed into simple fractions.

To do this, you need to find the roots of the equation:
Q n (x) = 0 .
Using the obtained roots, you need to represent the denominator as a product of factors:
Q n (x) = s (x-a) n a (x-b) n b ... (x 2 +ex+f) n e (x 2 +gx+k) n g ....
Here s is the coefficient for x n , x 2 + ex + f > 0 , x 2 + gx + k > 0 , ... .

After that, decompose the fraction into the simplest:

Integrating, we obtain an expression consisting of simpler integrals.
Integrals of the form

are reduced to tabular substitution t = x - a .

Consider the integral:

Let's transform the numerator:
.
Substituting into the integrand, we obtain an expression that includes two integrals:
,
.
First, substitution t \u003d x 2 + ex + f is reduced to a table.
The second, according to the reduction formula:

is reduced to the integral

We bring its denominator to the sum of squares:
.
Then by substitution, the integral

is also given in the table.

Integration of irrational functions

Let's introduce a notation. Let R(u 1 , u 2 , ... , u n) denote a rational function of the variables u 1 , u 2 , ... , u n . That is
,
where P, Q are polynomials in variables u 1 , u 2 , ... , u n .

Fractional linear irrationality

Consider integrals of the form:
,
Where - rational numbers, m 1 , n 1 , ..., m s , n s are integers.
Let n - common denominator numbers r 1 , ..., r s .
Then the integral is reduced to the integral of rational functions by substitution:
.

Integrals from differential binomials

Consider the integral:
,
where m, n, p are rational numbers, a, b are real numbers.
Such integrals reduce to integrals of rational functions in three cases.

1) If p is an integer. Substitution x = t N , where N is the common denominator of the fractions m and n .
2) If is an integer. Substitution a x n + b = t M , where M is the denominator of p .
3) If is an integer. Substitution a + b x - n = t M , where M is the denominator of p .

If none of the three numbers is an integer, then by Chebyshev's theorem integrals of this form cannot be expressed by a finite combination elementary functions.

In some cases, it may be useful to first reduce the integral to more convenient values of m and p . This can be done using the cast formulas:
;
.

Integrals Containing the Square Root of a Square Trinomial

Here we consider integrals of the form:
,

Euler substitutions

Such integrals can be reduced to integrals of rational functions of one of the three Euler substitutions:
, for a > 0 ;
, for c > 0 ;
, where x 1 is the root of the equation a x 2 + b x + c = 0. If this equation has real roots.

Trigonometric and hyperbolic substitutions

Direct Methods

In most cases, Euler substitutions result in longer computations than direct methods. Using direct methods, the integral is reduced to one of the following types.

I type

Integral of the form:
,
where P n (x) is a polynomial of degree n.

Such integrals are found by the method of indefinite coefficients, using the identity:

Differentiating this equation and equating the left and right sides, we find the coefficients A i .

II type

Integral of the form:
,
where P m (x) is a polynomial of degree m.

Substitution t = (x - α) -1 this integral is reduced to the previous type. If m ≥ n, then the fraction should have an integer part.

III type

The third and most difficult type:
.

Here you need to make a substitution:
.
Then the integral will take the form:
.
Further, the constants α, β must be chosen such that the coefficients at t vanish:
B = 0, B 1 = 0 .
Then the integral decomposes into the sum of integrals of two types:
;
,
which are integrated, respectively, by substitutions:
z 2 \u003d A 1 t 2 + C 1;
y 2 \u003d A 1 + C 1 t -2.

General case

Integration of transcendental (trigonometric and exponential) functions

We note in advance that those methods that are applicable to trigonometric functions, are also applicable to hyperbolic functions. For this reason, we will not consider the integration of hyperbolic functions separately.

Integration of rational trigonometric functions of cos x and sin x

Consider integrals of trigonometric functions of the form:
,
where R is a rational function. This may also include tangents and cotangents, which should be converted through sines and cosines.

When integrating such functions, it is useful to keep in mind three rules:
1) if R( cosx, sinx) multiplied by -1 from the sign change in front of one of the quantities cos x or sin x, then it is useful to denote the other of them by t .
2) if R( cosx, sinx) does not change from changing sign at the same time before cos x And sin x, then it is useful to put tan x = t or ctg x = t.
3) substitution in all cases leads to an integral of a rational fraction. Unfortunately, this substitution results in longer calculations than the previous ones, if applicable.

Product of power functions of cos x and sin x

Consider integrals of the form:

If m and n are rational numbers, then one of the permutations t = sin x or t= cos x the integral reduces to the integral of the differential binomial.

If m and n are integers, then the integrals are calculated by integrating by parts. This results in the following reduction formulas:

;
;
;
.

Integration by parts

Application of the Euler formula

If the integrand is linear with respect to one of the functions
cos ax or sinax, then it is convenient to apply the Euler formula:
e iax = cos ax + isin ax(where i 2 = - 1 ),
replacing this function with eiax and highlighting the real (when replacing cos ax) or the imaginary part (when replacing sinax) from the result.

References:
N.M. Gunther, R.O. Kuzmin, Collection of problems in higher mathematics, Lan, 2003.

Definition of the indefinite integral

The whole set of antiderivatives of the function f (x) can be called the indefinite integral of this function. With this in mind, the formula will look like ∫ f (x) d x = F (x) + C . In this case, the expression f (x) d x is an integrand, and f (x) is an integrand. The integrand is the differential of the function f(x).

Given a function differential, we can find the unknown function.

The result of indefinite integration will not be one function F (x) , but the set of its antiderivatives F (x) + C .

Knowing the properties of the derivative, we can formulate and prove the properties of the indefinite integral (properties of the antiderivative).

∫ f(x)dx" = F(x) + C" = f(x)

The derivative of the integration result is equal to the integrand.

∫ d(F(x)) = ∫ F"(x) d x = ∫ f(x) d x = F(x) + C

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

∫ k · f (x) d x = k · ∫ f (x) d x , where k is an arbitrary constant. The coefficient can be taken out of the sign of the indefinite integral.

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

∫ f (x) ± g (x)) d x = ∫ f (x) d x ± ∫ g (x) d x

We gave intermediate equalities of the first and second properties of the indefinite integral as an explanation.

In order to prove the third and fourth properties, it is necessary to find the derivatives of the right-hand sides of the equalities:

k ∫ f (x) d x " = k ∫ d (x) d x " = k f (x) ∫ f (x) d x ± ∫ g (x) d x " = ∫ f (x) d x " ± ∫ g (x) d x " = f (x) ± g (x)

The derivatives of the right-hand sides of the equalities are equal to the integrands, which is a proof of the first property. We use it in the last transitions.

As you can see, the integration problem is an inverse process with respect to the differentiation problem. Both of these tasks are closely related.

The first property can be used to perform an integration test. To check, it is enough for us to calculate the derivative of the result obtained. If the resulting function is equal to the integrand, then the integration is carried out correctly.

Thanks to the second property, given the known differential of a function, we can find its antiderivative and use it to calculate the indefinite integral.

Consider an example.

Example 1

Let's find the antiderivative of the function f (x) = 1 x, the value of which is equal to one at x = 1.

Solution

Using the table of derivatives of the main elementary functions, we obtain

d (ln x) = (ln x) " d x = d x x = f (x) d x ∫ f (x) d x = ∫ d x x = ∫ d (ln (x))

Using the second property ∫ d (ln (x)) = ln (x) + C , we get the set of antiderivatives ln (x) + C . For x = 1, we get the value ln (1) + C = 0 + C = C. According to the condition of the problem, this value should be equal to one, therefore, С = 1. The desired antiderivative will take the form ln (x) + 1 .

Answer: f(x) = 1 x = log(x) + 1

Example 2

It is necessary to find the indefinite integral ∫ 2 sin x 2 cos x 2 d x and check the result of the calculation by differentiation.

Solution

We use the formula for the sine of a double angle from the trigonometry course 2 sin x 2 cos x 2 \u003d sin x for calculations, we get ∫ 2 sin x 2 cos x 2 d x \u003d ∫ sin x d x.

We use the table of derivatives for trigonometric functions, we get:

d (cos x) = cos x " d x = - sin x d x ⇒ sin x d x = - d (cos x)

That is, ∫ sin x d x = ∫ (- d (cos x))

Using the third property of the indefinite integral, we can write ∫ - d (cos x) = - ∫ d (cos x) .

By the second property, we get - ∫ d (cos x) = - (cos x + C)

Therefore, ∫ 2 sin x 2 cos x 2 d x = - cos x - C .

Let's check the obtained result by differentiation.

Let's differentiate the resulting expression:
- cos x - C "= - (cos x)" - (C) "= - (- sin x) = sin x = 2 sin x 2 cos x 2

As a result of the verification, we have obtained an integrand. This means that the integration was carried out by us correctly. To implement the last transition, we used the formula for the sine of a double angle.

Answer:∫ 2 sin x 2 cos x 2 d x = - cos x - C

If the table of derivatives of the basic elementary functions is rewritten in the form of differentials, then from it, according to the second property of the indefinite integral, it is possible to compile a table of antiderivatives.

We will consider this topic in more detail in the next section "Table of antiderivatives (table of indefinite integrals)".

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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), that is, this 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) , 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 (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 problems of finding an 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

Solution. 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 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:

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, 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 a 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 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.