What is called an indefinite integral. Antiderivative and indefinite integral, their properties. Basic integration techniques

Definition of antiderivative.

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

If we take into account the fact that the derivative of the constant C is equal to zero, then the equality is true . 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.

Definition of an indefinite integral.

The entire set of antiderivatives of the function f(x) is called the indefinite integral of this function and is denoted .

The expression is called integrand, and f(x) – integrand function. The integrand represents the differential of the function f(x) .

The action of finding an unknown function given its differential is called uncertain integration, because the result of integration is not one function F(x), but a set of its antiderivatives F(x)+C.

Based on the properties of the derivative, one can formulate and prove properties of the indefinite integral(properties of an antiderivative).

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 a proof due to the first property. It is also used in the last transitions.

Thus, the problem of integration is the inverse of the problem of differentiation, and there is a very close connection between these problems:

• the first property allows one to check 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, this will mean that the integration was carried out correctly;
• the second property of the indefinite integral allows one to find its antiderivative from a known differential of a function. The direct calculation of indefinite integrals is based on this property.

Let's look at an example.

Example.

Find the antiderivative of the function whose value is equal to one at x = 1.

Solution.

We know from differential calculus that (just look at the table of derivatives of basic elementary functions). Thus, . By the second property . That is, we have many antiderivatives. For x = 1 we get the value . According to the condition, this value must be equal to one, therefore, C = 1. The desired antiderivative will take the form .

Example.

Find the indefinite integral and check the result by differentiation.

Solution.

Using the double angle sine formula from trigonometry , That's why

The concept of an indefinite integral. differentiation is the action by which, given a given function, its derivative or differential is found. For example, if F(x) = x 10, then F" (x) = 10x 9, dF (x) = 10x 9 dx.

Integration - This is the opposite of differentiation. Using integration over a given derivative or differential of a function, the function itself is found. For example, if F" (x) = 7x 6, then F (x) == x 7, since (x 7)" = 7x 6.

Differentiable function F(x), xЄ]a; b[ is called antiderivative for the function f (x) on the interval ]а; b[, if F" (x) = f (x) for each xЄ]a; b[.

Thus, for the function f(x) = 1/cos 3 x the antiderivative is the function F(x)= tan x, since (tg x)"= 1/cos 2 x.

The set of all antiderivative functions f(x) on the interval ]а; b[ is called indefinite integral from the function f(x) on this interval and write f (x)dx = F(x) + C. Here f(x)dx is the integrand;

F(x)-integral function; x-variable of integration: C is an arbitrary constant.

For example, 5x 4 dx = x 5 + C, since (x 3 + C)" = 5x 4.

Let's give basic properties of the indefinite integral. 1. The differential of the indefinite integral is equal to the integrand:

D f(x)dx=f(x)dx.

2. The indefinite integral of the differential of a function is equal to this function added to an arbitrary constant, i.e.

3. The constant factor can be taken out of the sign of the indefinite integral:

af(x)dx = a f(x)dx

4. The indefinite integral of the algebraic sum of functions is equal to the algebraic sum of the indefinite integrals of each function:

(f 1 (x) ±f 2 (x))dx = f 1 (x)dx ± f 2 (x)dx.

Basic integration formulas

(tabular integrals).

6.

Example 1. Find

Solution. Let's make the substitution 2 - 3x 2 = t then -6xdx =dt, xdx = -(1/6)dt. Next, we get

Example 3. Find

Solution. Let's put 10x = t; then 10dx = dt, whence dx=(1/10)dt.

3.

So, when finding sinl0xdx, you can use the formula sinkxdx = - (1/k) cos kx+C, where k=10.

Then sinl0xdx = -(1/10) сos10х+С.

Self-test questions and exercises

1. What action is called integration?

2. What function is called the antiderivative for the function f(x)?

3. Define an indefinite integral.

4. List the main properties of the indefinite integral.

5. How can you check integration?

6. Write the basic integration formulas (tabular integrals).

7. Find the integrals: a) b) c)

where a is the lower limit, b is the upper limit, F (x) is some antiderivative of the function f (x).

From this formula one can see the procedure for calculating a certain integral: 1) find one of the antiderivatives F (x) of a given function; 2) find the value of F (x) for x = a and x = b; 3) calculate the difference F (b) - F (a).

Example 1. Calculate integral

Solution. Let's use the definition of a power with a fractional and negative exponent and calculate the definite integral:

2. The integration segment can be divided into parts:

3. The constant factor can be taken out of the integral sign:

4. The integral of the sum of functions is equal to the sum of the integrals of all terms:

2) Let us determine the limits of integration for the variable t. For x=1 we get tn =1 3 +2=3, for x=2 we get tb =2 3 +2=10.

Example 3. Calculate integral

Solution. 1) put cos x=t; then – sinxdx =dt and

sinxdx = -dt. 2) Let us determine the limits of integration for the variable t: t n =cos0=1:t in =cos (π/2)=0.

3) Expressing the integrand in terms of t and dt and moving to new limits, we obtain

Let's calculate each integral separately:

Example 5. Calculate the area of ​​the figure bounded by the parabola y = x 2, straight lines x = - 1, x = 2 and the abscissa axis (Fig. 47).

Solution. Applying formula (1), we obtain

those. S=3 sq. units

The area of ​​the ABCD figure (Fig. 48), limited by the graphs of continuous functions y = f 1 (x) and y f 2 = (x), where x Є[a, b], line segments x = a and x = b, is calculated by formula

		The volume of a body formed by rotation around the Oy axis of a curvilinear trapezoid aAB, bounded by a continuous curve x=f(y), where Є [a, b], segment [a, b] of the Oy axis, line segments y = a and y = b ( Fig. 53), calculated by the formula Path taken by a point. If a point moves rectilinearly and its speed v=f(t) is a known function of time t, then the path traveled by the point in a period of time is calculated by the formula Self-test questions 1. Give the definition of a definite integral. 2. List the main properties of the definite integral. 3. What is the geometric meaning of a definite integral? 4. Write formulas to determine the area of ​​a plane figure using a definite integral. 5. What formulas are used to find the volume of a body of rotation? 6. Write a formula to calculate the distance traveled by the body. 7. Write a formula to calculate the work done by a variable force. 8. What formula is used to calculate the force of liquid pressure on a plate?

A function that can be restored from its derivative or differential is called antiderivative.

Definition. Function F(x) called antiderivative for function

f(x) on some interval, if at each point of this interval

F"(x) = f(x)

or, which is also,

dF(x) = f(x)dx

For example, F(x) = sin x is an antiderivative for f(x) = cos x on the entire number line OX, because

(sin x)" = cos x

If the function F(x) there is an antiderivative for the function f(x) on [ a; b], then the function F(x) + C, Where C any real number is also antiderivative for f(x) at any value C. Really ( F(x) + C)" = F"(x) + C" = f(x).

Example.

Definition. If F(x) one of the antiderivatives for the function f(x) on [ a; b], then the expression F(x) + C, Where C an arbitrary constant called indefinite integral from function f(x) and is denoted by the symbol ʃ f(x)dx(read: indefinite integral of f(x) on dx). So,

ʃ f (x ) dx = F (x ) +C ,

Where f(x) called the integrand function, f(x)dx- integrand expression, x is the variable of integration, and the symbol ʃ is the sign of the indefinite integral.

Properties of the indefinite integral and its geometric properties.

From the definition of the indefinite integral it follows that:

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

Really, F"(x) = f(x) and ʃ f(x)dx = F(x)+C. Then

2. The differential of the indefinite integral is equal to the integrand

Really,

3. The indefinite integral of the derivative is equal to the function itself plus an arbitrary constant:

Really, F"(x) = f(x). Then,

4. The indefinite integral of the differential is equal to the differentiable function plus an arbitrary constant:

Really, . Then,

5. Constant multiplier k(k≠ 0) can be taken out as the sign of the indefinite integral:

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

Let's call the graph an antiderivative F(x) of the integral curve. Graph of any other antiderivative F(x) + C obtained by parallel transfer of the integral curve F(x) along the axis OY.

Example.

Table of basic integrals

Basic integration techniques

1. Direct (tabular) integration.

Direct (tabular) integration is the reduction of the integral to tabular form using the basic properties and formulas of elementary mathematics.

Example 1.

Solution:

Example2 .

Solution:

Example3 .

Solution:

2. Method of bringing under the differential.

Example 1.

Solution:

Example2 .

Solution:

Example3 .

Solution:

Example4 .

Solution:

Example5 .

Solution:

Example6 .

Solution:

Example7 .

Solution:

Example8 .

Solution:

Example9 .

Solution:

Example10 .

Solution:

3. The second method of connecting to the differential.

Example 1.

Solution:

Example2 .

Solution:

4. Variable replacement (substitution) method.

Example.

Solution:

5. Method of integration by parts.

Using this formula, the following types of integrals are taken:

1 type

, formula applies n- once, the rest dv.

2 type.

, The formula is applied once.

Example1 .

Solution:

Example 2.

Solution:

Example3 .

Solution:

Example4 .

Solution:

INTEGRATION OF RATIONAL FRACTIONS.

A rational fraction is the ratio of two polynomials - degrees m and - degrees n,

The following cases are possible:

1. If , then use the angle division method to eliminate the whole part.

2. If the denominator also has a square trinomial, then the method of addition to a perfect square is used.

Example 1.

Solution:

Example2 .

Solution:

3. The method of indefinite coefficients when decomposing a proper rational fraction into a sum of simple fractions.

Any proper rational fraction, where, can be represented as a sum of simple fractions:

Where A, B, C, D, E, F, M, N,…‒ uncertain coefficients.

To find the uncertain coefficients, the right-hand side must be reduced to a common denominator. Since the denominator coincides with the denominator of the fraction on the right side, they can be discarded and the numerators can be equated. Then, equating the coefficients at the same degrees x on the left and right sides, we obtain a system of linear equations with n- unknown. Having solved this system, we find the required coefficients A, B, C, D and so on. And, therefore, we will decompose a proper rational fraction into simpler fractions.

Let's look at possible options using examples:

1. If the denominator factors are linear and different:

2. If among the denominator factors there are short factors:

3. If among the factors of the denominator there is a square trinomial that cannot be factorized:

Examples: Decompose a rational fraction into the sum of the simplest ones. Integrate.

Example 1.

Since the denominators of the fractions are equal, the numerators must also be equal, i.e.

Example 2.

Example3 .

Lesson 2. Integral calculus

The indefinite integral and its geometric meaning. Basic properties of the indefinite integral.

Basic methods for integrating the indefinite integral.

Definite integral and its geometric meaning.

Newton-Leibniz formula. Methods for calculating the definite integral.

Knowing the derivative or differential of a function, you can find the function itself (restore the function). This action, the inverse of differentiation, is called integration.

Antiderivative function in relation to a given function the following function is called
, the derivative of which is equal to the given function, i.e.

For this function There are an infinite number of antiderivative functions, because any of the functions
, is also an antiderivative of .

The set of all antiderivatives for a given function is called its indefinite integral is indicated by the symbol:

, Where

called the integrand, the function
- integrand function.

Geometric meaning of the indefinite integral. Geometrically, an indefinite integral is a family of integral curves on a plane obtained by parallel transfer of the graph of a function
along the ordinate axis (Fig. 3).

Basic properties of the indefinite integral

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

Property 2. The differential of an indefinite integral is equal to the integrand:

Property 3. The integral of the differential of a function is equal to this function plus const:

Property 4. Linearity of the integral.

Table of basic integrals

	Integral
power
indicative
trigonometric
reverse trigonometric

Basic integration methods

Method of integration by parts is a method that involves using the formula:

.

This method is used if the integral
is easier to solve than
. As a rule, this method solves integrals of the form
, Where
is a polynomial, and is one of the following functions:
,
,
, , ,
,
.

Let's consider some function
, defined on the interval
, rice. 4. Let's perform 5 operations.

1. Let's divide the interval with points in an arbitrary manner into parts. Let's denote
, and the largest of the lengths of these partial sections will be denoted by , we will call it the crushing rank.

2. On each partial plot
let's take an arbitrary point and calculate the value of the function in it
.

3. Let's compose a work

4. Let's make a sum
. This sum is called the integral sum or Riemann sum.

5. By reducing crushing (by increasing the number of crushing points) and at the same time directing the crushing rank to zero (
) i.e. (by increasing the number of crushing points, we make sure that the length of all partial sections decreases and tends to zero
), we will find the limit of the sequence of integral sums

If this limit exists and does not depend on the method of division and choice of points, then it is called definite integral from a function over an interval and is denoted as follows:
.

Geometric meaning of a definite integral. Let us assume that the function is continuous and positive on the interval. Consider a curved trapezoid ABCD(Fig. 4). Cumulative sum
gives us the sum of the areas of rectangles with bases
and heights
. It can be taken as an approximate value of the area of ​​a curved trapezoid ABCD , i.e.

,

Moreover, this equality will be the more accurate, the finer the crushing, and in the limit at n→+∞ And λ → 0 we will get:

.

This is the geometric meaning of the definite integral.

Basic properties of the definite integral

Property 1. A definite integral with equal limits is equal to zero.

Property 2. When the limits of integration are swapped, the definite integral changes sign to the opposite one.

Property 3. Linearity of the integral.

Property 4. Whatever the numbers are, if the function
integrable on each interval
,
,
(Fig. 5), then:

Theorem. If a function is continuous on the interval, then the definite integral of this function over the interval is equal to the difference in the values ​​of any antiderivative of this function at the upper and lower limits of integration, i.e.

(Newton-Leibniz formula) .

This formula reduces finding definite integrals to finding indefinite integrals. Difference
is called the increment of the antiderivative and is denoted
.

Let's consider the main ways to calculate a definite integral: change of variables (substitution) and integration by parts.

Substitution (change of variable) in a definite integral - you need to do the following:

And
;

Comment. When evaluating definite integrals using substitution, there is no need to return to the original argument.

2. Integration by parts in a definite integral comes down to using the formula:

.

Examples of problem solving

Exercise 1. Find the indefinite integral by direct integration.

1.
. Using the property of the indefinite integral, we take a constant factor beyond the sign of the integral. Then, performing elementary mathematical transformations, we reduce the integrand function to power form:

.

Task 2. Find the indefinite integral using the change of variable method.

1.
. Let's make a variable change
, Then . The original integral will take the form:

Thus, we have obtained an indefinite integral of a tabular form: a power function. Using the rule for finding the indefinite integral of a power function, we find:

Having made the reverse substitution, we get the final answer:

Task 3. Find the indefinite integral using the method of integration by parts.

1.
. Let us introduce the following notation: meaning ... basic concept integral calculus– concept uncertain integral ... uncertain integral Basic properties uncertain integral Use table main uncertain ...

Work program of the academic discipline "higher mathematics" Cycle

Working programm

... basic laws... Integral calculus functions of one variable Antiderivative. Uncertain integral And his properties ... integral And his geometric meaning. Integral... coordinates. Uncertain integral and... and practical classes". Petrushko I.M., ...

​Integral is an important part of differential calculus. Integrals can be double, triple, etc. To find the surface area and volume of geometric bodies, various types of integrals are used.

The indefinite integral has the form: \(∫f (x)\, dx\) and the definite integral has the form: \(\int_a^b \! f (x)\, dx\)

The region of the plane limited by the graph of the definite integral:

Integration operations are the inverse of differentiation. For this reason, we need to remember the antiderivative, function, table of derivatives.

The function \(F (x) = x^2\) is an antiderivative of the function \(f (x) = 2x\) . The functions \(f (x) = x^2+2\) and \(f (x) = x^2+7\) are also antiderivatives for the function \(f (x) = 2x\). \(2\) and \(7-\) are constants whose derivatives are equal to zero, so we can substitute them as much as we like, the value of the antiderivative will not change. To write an indefinite integral, use the sign \(∫\) . Indefinite integral is the set of all antiderivatives of the function \(f (x) = 2x\). Integration operations are the inverse of differentiation. \(∫2x = x^2+C\) , where \(C\) is the constant of integration, that is, if we calculate the derivative \(x^2\) , we get \(2x\) , and this is \ (∫2x\) . Easy, isn't it? If you don't understand, then you need to repeat the derivative of the function. Now we can derive the formula by which we will calculate the integral: \(∫u^ndu=\frac(u^n+1) (n+1), n ​​≠ -1\). we subtracted 1, now we add 1, n cannot be equal to 0. There are also other integration rules for other basic functions that need to be learned:

Solving an indefinite integral is the inverse process of finding antiderivatives of a differential equation. We find a function whose derivative is an integral, and don't forget to add "+ C" at the end.

The principles of integral calculus were formulated independently by Isaac Newton and Gottfried Leibniz at the end of the 17th century. Bernhard Riemann gave a strict mathematical definition of integrals. The first documented systematic method capable of determining integrals is the calculus method of the ancient Greek astronomer Eudoxus, who tried to find areas and volumes by breaking them down into an infinite number of known areas and volumes. This method was further developed and used by Archimedes in the 3rd century BC. e. and was used to calculate the areas of parabolas and approximate the area of ​​a circle.

A similar method was independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of ​​a circle. This method was later used in the 5th century by Chinese father-and-son mathematicians ZU Chongzhi and ZU Geng to find the volume of a sphere.

The next significant advances in integral calculus did not appear until the 17th century. During this time, the work of Cavalieri and Fermat began to lay the foundations of modern calculus.

In particular, the fundamental theorem of integral calculus allows us to solve a much wider class of problems. Equally important is the complex mathematical framework that Newton and Leibniz developed. This structure of integrals is taken directly from Leibniz's work and became modern integral calculus. The calculus was modified by Riemann, using limits. Subsequently, more general functions were considered, especially in the context of Fourier analysis, to which Riemann's definition does not apply. Lebesgue formulated another definition of the integral, based in measure theory (a subfield of real analysis).

The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675.

Integrals are widely used in many areas of mathematics. For example, in probability theory, integrals are used to determine the probability of some random variable falling within a certain range.

Integrals can be used to calculate the area of ​​a two-dimensional region that has a curved boundary, as well as to calculate the volume of a three-dimensional object that has a curved boundary.

Integrals are used in physics, in fields such as kinematics, to find displacement, time, and velocity.