The function is called the antiderivative of the if function. Extracurricular lesson - Antiderivative. Integration. Integral computation rules for dummies

Antiderivative

Defining an Antiderivative Function

• Function y = F (x) called the antiderivative for the function y = f (x) at a given interval NS, if for all NS ∈NS equality holds: F ′ (x) = f (x)

It can be read in two ways:

1. f derivative of a function F
2. F antiderivative for function f

The property of antiderivatives

• If F (x)- antiderivative for function f (x) on a given interval, then the function f (x) has infinitely many antiderivatives, and all these antiderivatives can be written as F (x) + C, where C is an arbitrary constant.

Geometric interpretation

• Graphs of all antiderivatives of a given function f (x) are obtained from the graph of any one antiderivative by parallel translations along the O axis at.

Antiderivatives calculation rules

1. The antiderivative of the sum is equal to the sum of the antiderivatives... If F (x)- antiderivative for f (x), and G (x) is the antiderivative for g (x), then F (x) + G (x)- antiderivative for f (x) + g (x).
2. The constant factor can be moved outside the sign of the derivative... If F (x)- antiderivative for f (x), and k- constant, then k F (x)- antiderivative for k f (x).
3. If F (x)- antiderivative for f (x), and k, b- permanent, moreover k ≠ 0, then 1 / k F (kx + b)- antiderivative for f (kx + b).

Remember!

Any function F (x) = x 2 + C , where C is an arbitrary constant, and only such a function is the antiderivative for the function f (x) = 2x.

• For example:

  F "(x) = (x 2 + 1)" = 2x = f (x);

  f (x) = 2x, since F "(x) = (x 2 - 1)" = 2x = f (x);

  f (x) = 2x, since F "(x) = (х 2 –3)" = 2x = f (x);

The relationship between the graphs of a function and its antiderivative:

1. If the graph of the function f (x)> 0 F (x) increases in this interval.
2. If the graph of the function f (x)<0 on the interval, then the graph of its antiderivative F (x) decreases in this interval.
3. If f (x) = 0, then the graph of its antiderivative F (x) at this point changes from increasing to decreasing (or vice versa).

To denote the antiderivative, the sign of an indefinite integral is used, that is, an integral without indicating the limits of integration.

Indefinite integral

Definition:

• An indefinite integral of a function f (x) is the expression F (x) + C, that is, the collection of all antiderivatives of a given function f (x). The indefinite integral is denoted as follows: \ int f (x) dx = F (x) + C

• f (x)- called the integrand;
• f (x) dx- called the integrand;
• x- called the variable of integration;
• F (x)- one of the antiderivatives of the function f (x);
• WITH is an arbitrary constant.

Indefinite integral properties

1. The derivative of the indefinite integral is equal to the integrand: (\ int f (x) dx) \ prime = f (x).
2. The constant factor of the integrand can be taken outside the integral sign: \ int k \ cdot f (x) dx = k \ cdot \ int f (x) dx.
3. The integral of the sum (difference) of functions is equal to the sum (difference) of the integrals of these functions: \ int (f (x) \ pm g (x)) dx = \ int f (x) dx \ pm \ int g (x) dx.
4. If k, b are constants, and k ≠ 0, then \ int f (kx + b) dx = \ frac (1) (k) \ cdot F (kx + b) + C.

Table of antiderivatives and indefinite integrals

Function f (x)	Antiderivative F (x) + C	Indefinite integrals \ int f (x) dx = F (x) + C
0	C	\ int 0 dx = C
f (x) = k	F (x) = kx + C	\ int kdx = kx + C
f (x) = x ^ m, m \ not = -1	F (x) = \ frac (x ^ (m + 1)) (m + 1) + C	\ int x (^ m) dx = \ frac (x ^ (m + 1)) (m + 1) + C
f (x) = \ frac (1) (x)	F (x) = l n \ lvert x \ rvert + C	\ int \ frac (dx) (x) = l n \ lvert x \ rvert + C
f (x) = e ^ x	F (x) = e ^ x + C	\ int e (^ x) dx = e ^ x + C
f (x) = a ^ x	F (x) = \ frac (a ^ x) (l na) + C	\ int a (^ x) dx = \ frac (a ^ x) (l na) + C
f (x) = \ sin x	F (x) = - \ cos x + C	\ int \ sin x dx = - \ cos x + C
f (x) = \ cos x	F (x) = \ sin x + C	\ int \ cos x dx = \ sin x + C
f (x) = \ frac (1) (\ sin (^ 2) x)	F (x) = - \ ctg x + C	\ int \ frac (dx) (\ sin (^ 2) x) = - \ ctg x + C
f (x) = \ frac (1) (\ cos (^ 2) x)	F (x) = \ tg x + C	\ int \ frac (dx) (\ sin (^ 2) x) = \ tg x + C
f (x) = \ sqrt (x)	F (x) = \ frac (2x \ sqrt (x)) (3) + C
f (x) = \ frac (1) (\ sqrt (x))	F (x) = 2 \ sqrt (x) + C
f (x) = \ frac (1) (\ sqrt (1-x ^ 2))	F (x) = \ arcsin x + C	\ int \ frac (dx) (\ sqrt (1-x ^ 2)) = \ arcsin x + C
f (x) = \ frac (1) (\ sqrt (1 + x ^ 2))	F (x) = \ arctan x + C	\ int \ frac (dx) (\ sqrt (1 + x ^ 2)) = \ arctg x + C
f (x) = \ frac (1) (\ sqrt (a ^ 2-x ^ 2))	F (x) = \ arcsin \ frac (x) (a) + C	\ int \ frac (dx) (\ sqrt (a ^ 2-x ^ 2)) = \ arcsin \ frac (x) (a) + C
f (x) = \ frac (1) (\ sqrt (a ^ 2 + x ^ 2))	F (x) = \ arctg \ frac (x) (a) + C	\ int \ frac (dx) (\ sqrt (a ^ 2 + x ^ 2)) = \ frac (1) (a) \ arctg \ frac (x) (a) + C
f (x) = \ frac (1) (1 + x ^ 2)	F (x) = \ arctg + C	\ int \ frac (dx) (1 + x ^ 2) = \ arctg + C
f (x) = \ frac (1) (\ sqrt (x ^ 2-a ^ 2)) (a \ not = 0)	F (x) = \ frac (1) (2a) l n \ lvert \ frac (x-a) (x + a) \ rvert + C	\ int \ frac (dx) (\ sqrt (x ^ 2-a ^ 2)) = \ frac (1) (2a) l n \ lvert \ frac (x-a) (x + a) \ rvert + C
f (x) = \ tg x	F (x) = - l n \ lvert \ cos x \ rvert + C	\ int \ tg x dx = - l n \ lvert \ cos x \ rvert + C
f (x) = \ ctg x	F (x) = l n \ lvert \ sin x \ rvert + C	\ int \ ctg x dx = l n \ lvert \ sin x \ rvert + C
f (x) = \ frac (1) (\ sin x)	F (x) = l n \ lvert \ tg \ frac (x) (2) \ rvert + C	\ int \ frac (dx) (\ sin x) = l n \ lvert \ tg \ frac (x) (2) \ rvert + C
f (x) = \ frac (1) (\ cos x)	F (x) = l n \ lvert \ tg (\ frac (x) (2) + \ frac (\ pi) (4)) \ rvert + C	\ int \ frac (dx) (\ cos x) = l n \ lvert \ tg (\ frac (x) (2) + \ frac (\ pi) (4)) \ rvert + C

Newton-Leibniz formula

Let be f (x) given function, F its arbitrary antiderivative.

\ int_ (a) ^ (b) f (x) dx = F (x) | _ (a) ^ (b)= F (b) - F (a)

where F (x)- antiderivative for f (x)

That is, the integral of the function f (x) on the interval is equal to the difference of the antiderivatives at the points b and a.

Curved trapezoid area

Curved trapezoid is called a figure bounded by the graph of a non-negative and continuous function on a segment f, the Ox axis and straight lines x = a and x = b.

The area of ​​a curved trapezoid is found by the Newton-Leibniz formula:

S = \ int_ (a) ^ (b) f (x) dx

Definition of antiderivative.

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.

Definition of an indefinite integral.

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 integrand, and f (x) - integrand... The integrand is the differential of the function f (x).

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

Based on the properties of the derivative, it is possible to formulate and prove indefinite integral properties(properties of the 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 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.

Let's look at an example.

Example.

Find the antiderivative of a 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 a lot of antiderivatives. For x = 1, we get the value. By condition, this value should 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.

Double angle sine formula from trigonometry , therefore

Antiderivative function and indefinite integral

Fact 1. Integration is an action inverse to differentiation, namely, the restoration of a function from a known derivative of this function. The function thus restored F(x) is called antiderivative 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 of the function f(x) = cos x on the whole number line, since for any value of x (sin x) "= (cos x) .

Definition 2. The indefinite integral of a function f(x) is the set of all its antiderivatives... In this case, the record is used

∫

f(x)dx

,

where is the sign ∫ is called the integral sign, the function f(x) Is the integrand, and f(x)dx - an integrand.

So if F(x) Is some kind of antiderivative for f(x) , then

∫

f(x)dx = F(x) +C

where C - an 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 (traditional wooden door). Its function is to "be the door". And what is the door made of? Made of wood. 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 mean, for example, a tree species. Just like a door is made of wood with some tools, the derivative of a function is "made" from an antiderivative function using the formula that we learned by studying the derivative .

Then the table of functions of common objects and their corresponding antiderivatives ("to be a door" - "to be a tree", "to be a spoon" - "to be metal", etc.) is similar to the table of basic indefinite integrals, which will be given below. The table of indefinite integrals lists common functions with an indication of the antiderivatives from which these functions are "made". In the part of the problems of finding the indefinite integral, such integrands are given that, without special considerations, can be integrated directly, that is, according to the table of indefinite integrals. In more complicated problems, the integrand must first be transformed so that tabular integrals can be used.

Fact 2. When 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 various constants from 1 to infinity, you need to write a set of antiderivatives with an arbitrary constant C for example like this: 5 x³ + С. 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 differentiation 4 or 3, or any other constant vanish.

Let us pose the integration problem: for this 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 the 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, a function is an antiderivative for a function. However, it is not the only antiderivative for. They also serve as 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 number of antiderivatives that differ by a constant term. 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 NS, 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 next example, we are already referring to the table of integrals, which will be given in Section 3, after the properties of the indefinite integral. We do this before reading the entire table so that the essence of the above is clear. And after the table and properties, we will use them in the integration in their entirety.

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 entire table of indefinite integrals 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 by formula (7) at n= -1/4 find

The integral is not the function itself f, and its product by the differential dx... This is done primarily to indicate which variable is being searched for the antiderivative. For example,

, ;

here in both cases the integrand is equal, 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 the variable x, and in the second - as a function of z .

The process of finding the indefinite integral of a function is called the integration of this 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 angle of inclination of the tangent at each of its points is a given function f (x) abscissa of this point.

According to the geometric meaning of the derivative, the tangent of the angle of inclination of the tangent at a given point of the curve y = F (x) is equal to the value of the derivative F "(x)... Hence, we need to find such a function F (x), for which F "(x) = f (x)... Function required in the task F (x) is the antiderivative of f (x)... The condition of the problem is satisfied not by one curve, but by a family of curves. y = F (x) is 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) there 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.

Indefinite integral properties

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. 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 reciprocal operations.

Fact 6. Theorem 3. The constant factor in the integrand can be taken out of the indefinite integral sign , i.e.

Definition. The function F (x) is called the antiderivative for the function f (x) on the given interval, if for any x from the given interval F "(x) = f (x).

The main property of antiderivatives.

If F (x) is the antiderivative of the function f (x), then the function F (x) + C, where C is an arbitrary constant, is also the antiderivative of the function f (x) (i.e., all the antiderivatives of the function f (x) are written as F (x) + С).

Geometric interpretation.

The graphs of all antiderivatives of a given function f (x) are obtained from the graph of any one antiderivative by parallel translations along the Oy axis.

Antiderivatives table.

Rules for finding antiderivatives .

Let F (x) and G (x) be the antiderivatives of the functions f (x) and g (x), respectively. Then:

1.F ( x) ± G ( x) Is the antiderivative for f(x) ± g(x);

2. a F ( x) Is the antiderivative for af(x);

3. - antiderivative for af(kx +b).

Problems and tests on the topic "Antiderivative"

• Antiderivative

  Lessons: 1 Assignments: 11 Tests: 1

• Derivative and antiderivative - Preparation for the USE in mathematics USE in mathematics

  Tasks: 3

• Integral - Antiderivative and integral grade 11

  Lessons: 4 Assignments: 13 Tests: 1

• Calculating Areas Using Integrals - Antiderivative and integral grade 11

  Lessons: 1 Assignments: 10 Tests: 1

Having studied this topic, you should know what is called an antiderivative, its main property, geometric interpretation, the rules for finding antiderivatives; be able to find all antiderivatives of functions using the table and rules for finding antiderivatives, as well as the antiderivative passing through a given point. Let's consider solving problems on this topic using examples. Pay attention to the design of the solutions.

Examples.

1. Find out whether the function F ( x) = NS 3 – 3NS+ 1 antiderivative for function f(x) = 3(NS 2 – 1).

Solution: F "( x) = (NS 3 – 3NS+ 1) ′ = 3 NS 2 – 3 = 3(NS 2 – 1) = f(x), i.e. F "( x) = f(x), therefore, F (x) is an antiderivative for the function f (x).

2. Find all antiderivatives of the function f (x):

a) f(x) = NS 4 + 3NS 2 + 5

Solution: Using the table and the rules for finding antiderivatives, we get:

Answer:

b) f(x) = sin (3 x – 2)

Solution:

Primordial. A beautiful word.) First, a little Russian. This word is pronounced that way, and not "prototypical" as it might seem. Antiderivative is the basic concept of all integral calculus. Any integrals - indefinite, definite (you will get acquainted with them already in this semester), as well as double, triple, curvilinear, superficial (and these are the main characters of the second year) - are based on this key concept. It makes perfect sense to master. Go.)

Before getting acquainted with the concept of antiderivative, let's recall in the most general terms the most common derivative... Without delving into the boring theory of limits, argument increments and other things, we can say that finding the derivative (or differentiation) Is just a mathematical operation on function... And that's all. Any function is taken (for example, f (x) = x 2) and according to certain rules transforms into new function... And this one new function and called derivative.

In our case, before differentiation, there was a function f (x) = x 2, and after differentiation it became already another function f '(x) = 2x.

Derivative- because our new function f '(x) = 2x happened from function f (x) = x 2... As a result of the differentiation operation. And moreover, it is from her, and not from some other function ( x 3, for example).

Roughly speaking, f (x) = x 2- this is mom, and f '(x) = 2x- her beloved daughter.) This is understandable. Move on.

Mathematicians are restless people. They strive to find opposition for each of their actions. :) There is addition - there is also subtraction. There is multiplication - there is also division. Exponentiation is the extraction of the root. Sine is arcsine. Similarly, there is differentiation- so there is ... integration.)

And now we will pose such an interesting problem. For example, we have such a simple function f (x) = 1... And we need to answer this question:

The derivative of WHAT function gives us the functionf(x) = 1?

In other words, seeing a daughter, using DNA analysis, calculate who her mother is. :) So from what the original function (let's call it F (x)) our derivative function f (x) = 1? Or, in mathematical terms, for what function F (x), the following equality holds:

F '(x) = f (x) = 1?

An elementary example. I tried.) We just fit the function F (x) so that the equality works. :) Well, have you picked it up? Oh sure! F (x) = x. Because:

F '(x) = x' = 1 = f (x).

Found mom, of course F (x) = x must be called somehow, yes.) Meet!

Antiderivative for a functionf(x) such a function is calledF(x), whose derivative is equal tof(x), i.e. for which the equalityF’(x) = f(x).

That's all. No more scientific tricks. In the strict definition, an additional phrase is added "in the interval X"... But we will not go deep into these subtleties yet, because our primary task is to learn how to find these very primitives.

In our case, it turns out that the function F (x) = x is an antiderivative for function f (x) = 1.

Why? because F '(x) = f (x) = 1. The x-derivative is one. No objections.)

The term "primitive" in the philistine means "ancestor", "parent", "ancestor". Immediately, we remember the most dear and close person.) And the very search for the antiderivative is the restoration of the original function by its known derivative... In other words, this is an action reverse differentiation... And that's it! This fascinating process itself is also called quite scientific - integration... But about integrals- later. Patience, friends!)

Remember:

Integration is a mathematical operation on a function (just like differentiation).

Integration is the opposite of differentiation.

The antiderivative is the result of integration.

Now let's complicate the task. Let us now find the antiderivative for the function f (x) = x... That is, we will find such a function F (x) , to its derivative would equal x:

F '(x) = x

Someone who is friends with derivatives may come to mind something like:

(x 2) '= 2x.

Well, respect and respect to those who remember the table of derivatives!) Right. But there is one problem. Our original function f (x) = x, a (x 2) '= 2 x. Two X. And after differentiation, we should have just x... Not okay. But…

We are a scientific people. We got certificates.) And from school we know that both sides of any equality can be multiplied and divided by the same number (except for zero, of course)! So arranged. So we will realize this opportunity for our own good.)

We want a clean X to remain on the right, right? And two gets in the way ... So we take the ratio for the derivative (x 2) ’= 2x and divide both parts on this very two:

So, something is already being clarified. Move on. We know that any constant can be take out the sign of the derivative. Like this:

All formulas in mathematics work both from left to right and vice versa - from right to left. This means that, with the same success, any constant can be add under the derivative sign:

In our case, we hide the two in the denominator (or, which is the same, the coefficient 1/2) under the sign of the derivative:

And now attentively let's take a closer look at our record. What do we see? We see equality stating that the derivative of something(this is something- in brackets) equals x.

The resulting equality just means that the desired antiderivative for the function f (x) = x serves function F (x) = x 2/2 ... The one in parentheses under the stroke. Directly in the sense of the antiderivative.) Well, let's check the result. Let's find the derivative:

Fine! Received the original function f (x) = x... From what they danced, they returned to that. This means that our antiderivative was found correctly.)

What if f (x) = x 2? What is its antiderivative equal to? No problem! We all know (again, from the rules of differentiation) that:

3x 2 = (x 3) '

AND, that is,

Got it? Now we, imperceptibly for ourselves, have learned to count the primitives for any power function f (x) = x n... In the mind.) We take the initial indicator n, we increase it by one, and as compensation we divide the entire structure by n + 1:

The resulting formula, by the way, is valid not only for a natural indicator degree n, but also for any other - negative, fractional. This makes it easy to find antiderivatives from unpretentious fractions and roots.

For example:

Naturally, n ≠ -1 , otherwise the denominator of the formula turns out to be zero, and the formula loses its meaning.) About this special case n = -1 a little bit later.)

What is an indefinite integral? Integral table.

Let's say what is the derivative for the function F (x) = x? Well, one, one - I hear disgruntled answers ... That's right. Unit. But ... For the function G (x) = x + 1 derivative will also be equal to one:

Also, the derivative will be equal to one for the function x + 1234 , and for the function x-10 , and for any other function of the form x + C , where WITH - any constant. For the derivative of any constant is equal to zero, and from the addition / subtraction of zero, no one is cold or hot.)

It turns out ambiguity. It turns out that for the function f (x) = 1 the antiderivative is not only function F (x) = x but also the function F 1 (x) = x + 1234 and function F 2 (x) = x-10 etc!

Yes. That's right.) Any ( continuous in between) of the function, there is not just one antiderivative, but infinitely many - a whole family! Not just one mom or dad, but a whole pedigree, yeah.)

But! All our ancestral relatives have one important property in common. That's why they are relatives.) The property is so important that in the process of analyzing the methods of integration, we will recall it more than once. And we will remember for a long time.)

Here it is, this property:

Any two antiderivatives F 1 (x) andF 2 (x) from the same functionf(x) differ by a constant:

F 1 (x) - F 2 (x) = C.

Who cares about the proof - study the literature or lecture notes.) Okay, so be it, I will prove it. Fortunately, the proof here is elementary, in one action. Take equality

F 1 (x) - F 2 (x) = C

and we differentiate both parts of it. That is, we just stupidly put the strokes:

That's all. As they say, CHTD. :)

What does this property mean? And that two different antiderivatives from the same function f (x) cannot differ by some expression with x ... Only strictly constant! In other words, if we have a schedule of some kind one of the antiderivatives(let it be F (x)), then the graphs everyone else of our antiderivatives are constructed by parallel translation of the graph F (x) along the axis of the players.

Let's see how it looks with the example of the function f (x) = x... All its antiderivatives, as we already know, have the general form F (x) = x 2/2 + C ... In the picture it looks like infinite set of parabolas obtained from the "main" parabola y = x 2/2 by a shift along the OY axis up or down, depending on the value of the constant WITH.

Remember school function plotting y = f (x) + a shift schedule y = f (x) on "a" units along the axis of the players?) Here and then the same thing.)

Moreover, please note: our parabolas do not intersect anywhere! It is natural. After all, two different functions y 1 (x) and y 2 (x) will inevitably correspond two different values ​​of the constant – C 1 and C 2.

Therefore, the equation y 1 (x) = y 2 (x) never has solutions:

C 1 = C 2

x ∊ ∅ , because С 1 ≠ С2

And now we are smoothly approaching the second cornerstone concept of integral calculus. As we have just established, every function f (x) has an infinite set of antiderivatives F (x) + C, which differ from each other by a constant. This most infinite set also has its own special name.) Well, please love and favor!

What is an indefinite integral?

Set of all function antiderivatives f(x) is called indefinite integral from functionf(x).

That's the definition.)

"Uncertain" - because the set of all antiderivatives for the same function endlessly... There are too many different options.)

"Integral" - we will get acquainted with a detailed decoding of this atrocious word in the next large section devoted to certain integrals... In the meantime, in a rough form, we will consider the integral something common, single, whole... Integration - Union, generalization, in this case, the transition from the particular (derivative) to the general (antiderivative). Something like that.

The indefinite integral is denoted like this:

It is read in the same way as it is written: integral ff from x de x... Or integral from ff from x de x. Well, you get the idea.)

Now let's deal with the designations.

∫ - integral icon. The meaning is the same as the prime for the derivative.)

d - badgedifferential. We are not afraid! Why is it needed there - just below.

f (x) - integrand(through the "s").

f (x) dx - integrand expression. Or, roughly speaking, the "filling" of the integral.

According to the meaning of the indefinite integral,

Here F (x)- the same antiderivative for function f (x) we anyway found it ourselves. How exactly they found it is not the point. For example, we found that F (x) = x 2/2 for f (x) = x.

"WITH" - arbitrary constant. Or, more scientifically, integral constant... Or constant of integration. Everything is one.)

Now let's go back to our very first examples of finding an antiderivative. In terms of an indefinite integral, we can now safely write:

What is an integral constant and why is it needed?

The question is very interesting. And very (VERY!) Important. The integral constant from the whole infinite set of antiderivatives singles out that line, which passes through the given point.

What is the point. From the original infinite set of antiderivatives (i.e. indefinite integral) it is necessary to select the curve that will pass through the given point. With some specific coordinates. Such a task is always and everywhere encountered in the initial acquaintance with integrals. Both at school and at the university.

Typical problem:

Among the set of all antiderivatives of the function f = x, select the one that passes through the point (2; 2).

We start to think with our heads ... A lot of all primitives - this means, first you have to integrate our original function. That is, x (x). We dealt with this a little higher and received the following answer:

Now let's figure out what exactly we got. We got not one function, but a whole family of functions. Which ones? Species y = x 2/2 + C ... Depends on the value of the constant C. And now we have to "catch" this value of the constant.) Well, let's start catching?)

Our fishing rod - family of curves (parabolas) y = x 2/2 + C.

Constants - these are fish. Many-many. But each has its own hook and bait.)

What is the bait? Right! Our point (-2; 2).

So we substitute the coordinates of our point in the general view of the antiderivatives! We get:

y (2) = 2

From here it is already easy to search C = 0.

What does this mean? This means that of the entire infinite set of parabolas of the formy = x 2/2 + Conly parabola with constant C = 0 suits us! Namely:y = x 2/2. And only she. Only this parabola will pass through the point we need (-2; 2). And inall other parabolas from our family pass through this point will no longer be. Through some other points of the plane - yes, but through the point (2; 2) - no longer. Got it?

For clarity, here are two pictures for you - the whole family of parabolas (i.e. indefinite integral) and some specific parabola corresponding specific value of the constant and passing through specific point:

You see how important it is to consider the constant WITH when integrating! So do not neglect this letter "C" and do not forget to attribute it to the final answer.

And now let's figure out why the symbol is everywhere inside the integrals dx ... Students often forget about him ... And this, by the way, is also a mistake! And pretty rude. The point is that integration is the opposite of differentiation. And what exactly is the result of differentiation? Derivative? True, but not quite. Differential!

In our case, for the function f (x) differential of its antiderivative F (x), will:

Who does not understand this chain - urgently repeat the definition and meaning of the differential and how exactly it is revealed! Otherwise, you will mercilessly slow down in the integrals….

Let me remind you, in the roughest, philistine form, that the differential of any function f (x) is just a product f '(x) dx... And that's it! Take a derivative and multiply it on the differential of the argument(i.e. dx). That is, any differential, in fact, is reduced to calculating the usual derivative.

Therefore, strictly speaking, the integral is not "taken" from function f (x), as is commonly believed, but from differential f (x) dx! But, in a simplified version, it is customary to say that "the integral is taken from the function"... Or: "The function f is integrated(x)". This is the same. And we will say exactly the same. But about the icon dx we will not forget! :)

And now I will tell you how not to forget it when recording. Imagine first that you are calculating the usual derivative with respect to the variable x. How do you usually write it?

Like this: f ’(x), y’ (x), y ’x. Or more solidly, through the ratio of differentials: dy / dx. All these records show us that the derivative is taken with respect to x. And not by "game", "te" or some other variable there.)

The same is in integrals. Recording ∫ f (x) dx U.S. too as if shows that the integration is carried out precisely by variable x... Of course, this is all very simplified and crude, but it is understandable, I hope. And the odds forget ascribe the omnipresent dx decline sharply.)

So, what is the same indefinite integral - figured out. Great.) Now it would be nice to learn these same indefinite integrals calculate... Or, to put it simply, "take". :) And here two news awaits the students - good and not so good. Let's start with the good one for now.)

Good news. Integrals, as well as derivatives, have their own table. And all the integrals that we will meet along the way, even the most terrible and sophisticated, we according to certain rules we will somehow reduce to these very tabular.)

So here it is, table of integrals!

Here is such a beautiful table of integrals from the most popular functions. I recommend paying special attention to the group of formulas 1-2 (constant and power function). These are the most common formulas in integrals!

The third group of formulas (trigonometry), as you might guess, is obtained by simply inverting the corresponding formulas for the derivatives.

For example:

With the fourth group of formulas (exponential function) - everything is the same.

And here are the last four groups of formulas (5-8) for us new. Where did they come from and for what such merits did these exotic functions suddenly appear in the table of basic integrals? What makes these groups of functions stand out from the rest of the functions?

It so happened historically in the development process integration methods ... When we train to take the most diverse integrals, you will understand that integrals of the functions listed in the table are very, very common. So often that mathematicians classified them as tabular.) Many other integrals are expressed through them, from more complex constructions.

For the sake of interest, you can take some of these creepy formulas and differentiate. :) For example, the most brutal 7th formula.

Everything is fine. The mathematicians were not fooled. :)

It is desirable to know the table of integrals, like the table of derivatives, by heart. Anyway, the first four groups of formulas. This is not as difficult as it seems at first glance. Memorize the last four groups (with fractions and roots) while not worth it. All the same, at first you will be confused about where to write the logarithm, where is the arctangent, where is the arcsine, where 1 / a, where 1 / 2a ... There is only one way out - to solve more examples. Then the table by itself will gradually be remembered, and doubts will cease to gnaw.)

Particularly inquisitive persons, looking at the table, may ask: where in the table are the integrals of other elementary "school" functions - tangent, logarithm, "arches"? Let's say why there is an integral of the sine in the table, but there is NO, say, the integral of the tangent tg x? Or there is no integral of the logarithm ln x? From arcsine arcsin x? Why are they worse? But it is full of some kind of "left" functions - with roots, fractions, squares ...

Answer. Nothing worse.) Just the above integrals (from tangent, logarithm, arcsine, etc.) are not tabular ... And they are found in practice much less often than those presented in the table. Therefore to know by heart what they are equal to is not at all necessary. It's enough just to know how are they calculated.)

What, someone is still unbearable? So be it, especially for you!

Well, how are you going to memorize? :) Won't you? And don’t.) But don’t worry, we will definitely find all such integrals. In the relevant lessons. :)

Well, now we turn to the properties of the indefinite integral. Yes, there is nothing to be done! A new concept is introduced - right there and some of its properties are considered.

Indefinite integral properties.

Now is not good news.

Unlike differentiation, general standard integration rules fair for all occasions, in mathematics, no. It is fantastic!

For example, you all know perfectly well (I hope!) That any work any of two functions f (x) g (x) is differentiated like this:

(f (x) g (x)) ’= f’ (x) g (x) + f (x) g ’(x).

Any the quotient is differentiated like this:

And any complex function, no matter how twisted it may be, is differentiated like this:

And whatever functions are hidden under the letters f and g, the general rules will still work and the derivative, one way or another, will be found.

But with integrals, such a number will no longer work: for the product, the quotient (fraction), as well as a complex function of the general formulas of integration does not exist! There are no standard rules! Rather, they are. I shouldn't have offended mathematics.) But, firstly, there are much fewer of them than general rules for differentiation. And secondly, most of the integration methods that we will talk about in the next lessons are very, very specific. And they are valid only for a certain, very limited class of functions. Let's say just for fractional rational functions... Or something else.

And some integrals, although they exist in nature, are in no way expressed in terms of elementary "school" functions! Yes, there are plenty of such integrals! :)

That is why integration is much more time consuming and painstaking work than differentiation. But this has its own flavor. The lesson is creative and very exciting.) And if you master the table of integrals well and master at least two basic techniques, which we will talk about below (and), then you will really like the integration. :)

And now let's get acquainted, in fact, with the properties of the indefinite integral. They are nothing at all. Here they are.

The first two properties are completely analogous to those for derivatives and are called the linearity properties of the indefinite integral ... Everything is simple and logical here: the integral of the sum / difference is equal to the sum / difference of the integrals, and the constant factor can be taken outside the integral sign.

But the following three properties are fundamentally new for us. Let's take a closer look at them. They sound in Russian as follows.

Third property

The derivative of the integral is equal to the integrand

Everything is simple, like in a fairy tale. If we integrate the function, and then find the derivative of the result back, then ... we get the original integrand function. :) This property can always (and should) be used to check the final result of integration. Calculate the integral - differentiate the answer! Received the integrand function - OK. If they didn’t get it, it meant they messed it up somewhere. Look for the error.)

Of course, the answer can turn out to be such brutal and cumbersome functions that you are reluctant to differentiate them back and forth, yes. But it is better, if possible, to try to check yourself. At least in those examples where it's easy.)

Fourth property

The differential of the integral is equal to the integrand .

Nothing special here. The essence is the same, only dx appears at the end. According to the previous property and the rules for disclosing the differential.

Fifth property

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

Also a very simple property. We will also use it regularly in the process of solving integrals. Especially - in and.

These are the useful properties. I'm not going to be bored with their rigorous proofs here. I suggest those who wish to do it on their own. Directly within the meaning of the derivative and differential. I will only prove the last, fifth property, because it is less obvious.

So we have a statement:

We take out the "stuffing" of our integral and open it, according to the definition of the differential:

Just in case, I remind you that, according to our notation for the derivative and antiderivative, F’(x) = f(x) .

Now we insert our result back into the integral:

Received exactly indefinite integral definition (may the Russian language forgive me)! :)

That's all.)

Well. At this point, I consider our initial acquaintance with the mysterious world of integrals to have taken place. For today, I propose to round off. We are already armed enough to go on reconnaissance. If not a machine gun, then at least a water pistol with basic properties and a table. :) In the next lesson, we are already waiting for the simplest harmless examples of integrals for the direct application of the table and the written out properties.

See you!