The function is called primitive for a function if. An extracurricular lesson is primitive. Integration. Rules for calculating integrals for dummies

PRINTING

Definition of a primitive function

Function y \u003d f (x)called primitive for function y \u003d f (x) At a given interval X,if for all h. ∈ H. Equality is performed: F '(x) \u003d f (x)

You can read in two ways:

f. derived function F.
F. Perfect for function f.

Primitive property

If a F (x)- Perfect for function f (x) At a given gap, the function f (x) has infinitely many primitive, and all these primitive can be written as F (x) + withwhere C is an arbitrary constant.

Geometric interpretation

Graphs of all primitive this feature. f (x) obtained from the graph of any one primitive parallel transfer along the axis about w..

The rules for calculating the primary

The first amount is equal to the sum of the primordial. If a F (x) - Pred-like for f (x), and G (x) is a primitive for g (x)T. F (x) + g (x) - Pred-like for f (x) + g (x).
Permanent multiplier can be made for a derivative mark. If a F (x) - Pred-like for f (x), I. k. - constant, then k · f (x) - Pred-like for k · f (x).
If a F (x) - Pred-like for f (x), I. k, B. - constant, and k ≠ 0T. 1 / k · f (kx + b) - Pred-like for f (KX + B).

Remember!

Any feature F (x) \u003d x 2 + where C is an arbitrary constant, and only such a function is a primitive for function f (x) \u003d 2x.

For example:
F "(x) \u003d (x 2 + 1)" \u003d 2x \u003d f (x);

f (x) \u003d 2x, Because F "(x) \u003d (x 2 - 1)" \u003d 2x \u003d f (x);

f (x) \u003d 2x, Because F "(x) \u003d (x 2 -3)" \u003d 2x \u003d f (x);

The connection between the graphs of the function and its primary:

If the graph is function f (X)\u003e 0 F (x) Increases at this interval.
If the graph is function f (X)<0 on the interval, then the schedule is its primitive F (x) decreases at this interval.
If a f (x) \u003d 0then the graph of her primitive F (x) At this point changes with an increasing decrease (or vice versa).

To designate, the sign of an undefined integral is used, that is, the integral without specifying the integration limits.

Uncertain integral

Definition:

An uncertain integral from the function f (x) is the expression F (x) + C, that is, the combination of all the primary functions of the F (X). Denotes an indefinite integral as follows: \\ int f (x) dx \u003d f (x) + c

f (x)- refer to the integrated function;
f (x) dx- are called a concintive expression;
x. - call integration variable;
F (x) - one of the primitive functions f (x);
FROM - Arbitrary constant.

Properties of an indefinite integral

The derivative of an indefinite integral is equal to the integrand function: (\\ int f (x) dx) \\ prime \u003d f (x).
A permanent multiplier of the integrated expression can be made for an integral sign: \\ int k \\ cdot f (x) dx \u003d k \\ cdot \\ int f (x) dx.
The integral from the amount (difference) of functions is equal to the amount (difference) of the integrals from these functions: \\ int (f (x) \\ pm g (x)) dx \u003d \\ int f (x) dx \\ pm \\ int g (x) dx.
If a k, B.- constant, and k ≠ 0, then \\ int f (kx + b) dx \u003d \\ frac (1) (k) \\ cdot f (kx + b) + c.

Table of primary and uncertain integrals

Function f (X)	PRINTING F (x) + C	Uncertain integrals \\ int f (x) dx \u003d f (x) + c
0	C.	\\ int 0 dx \u003d c
f (x) \u003d k	F (x) \u003d kx + c	\\ int kdx \u003d kx + c
f (x) \u003d x ^ m, m \\ not \u003d -1	F (x) \u003d \\ frac (x ^ (m + 1)) (m + 1) + c	\\ int x (^ m) dx \u003d \\ frac (x ^ (m + 1)) (M + 1) + C
f (x) \u003d \\ FRAC (1) (X)	F (x) \u003d L n \\ LVERT X \\ RVERT + C	\\ int \\ FRAC (DX) (X) \u003d L N \\ LVERT X \\ RVERT + C
f (x) \u003d E ^ x	F (x) \u003d e ^ x + c	\\ int e (^ x) dx \u003d e ^ x + c
f (x) \u003d a ^ x	F (x) \u003d \\ frac (a ^ x) (L Na) + C	\\ int a (^ x) dx \u003d \\ frac (a ^ x) (L Na) + C
f (x) \u003d \\ sin x	F (x) \u003d - \\ cos x + c	\\ int \\ sin x dx \u003d - \\ cos x + c
f (x) \u003d \\ cos x	F (x) \u003d \\ sin x + c	\\ int \\ cos x dx \u003d \\ sin x + c
f (x) \u003d \\ FRAC (1) (\\ sin (^ 2) x)	F (x) \u003d - \\ ctg x + c	\\ int \\ FRAC (DX) (\\ sin (^ 2) x) \u003d - \\ CTG X + C
f (x) \u003d \\ FRAC (1) (\\ cos (^ 2) x)	F (x) \u003d \\ tg x + c	\\ int \\ FRAC (DX) (\\ sin (^ 2) x) \u003d \\ TG x + C
f (x) \u003d \\ sqrt (x)	F (x) \u003d \\ FRAC (2x \\ SQRT (X)) (3) + C
f (x) \u003d \\ FRAC (1) (\\ SQRT (X))	F (x) \u003d 2 \\ sqrt (x) + c
f (x) \u003d \\ FRAC (1) (\\ sqrt (1-x ^ 2))	F (x) \u003d \\ arcsin x + c	\\ int \\ FRAC (DX) (\\ SQRT (1-X ^ 2)) \u003d \\ ARCSIN X + C
f (x) \u003d \\ FRAC (1) (\\ sqrt (1 + x ^ 2))	F (x) \u003d \\ arctg x + c	\\ int \\ FRAC (DX) (\\ SQRT (1 + x ^ 2)) \u003d \\ arctg x + c
f (x) \u003d \\ FRAC (1) (\\ sqrt (a ^ 2-x ^ 2))	F (x) \u003d \\ arcsin \\ FRAC (X) (A) + C	\\ int \\ FRAC (DX) (\\ sqrt (a ^ 2-x ^ 2)) \u003d \\ arcsin \\ FRAC (X) (A) + C
f (x) \u003d \\ FRAC (1) (\\ sqrt (a ^ 2 + x ^ 2))	F (x) \u003d \\ arctg \\ FRAC (X) (A) + C	\\ int \\ FRAC (DX) (\\ SQRT (A ^ 2 + x ^ 2)) \u003d \\ FRAC (1) (a) \\ arctg \\ FRAC (X) (A) + C
f (x) \u003d \\ FRAC (1) (1 + x ^ 2)	F (x) \u003d \\ arctg + c	\\ int \\ FRAC (DX) (1 + x ^ 2) \u003d \\ arctg + c
f (x) \u003d \\ FRAC (1) (\\ sqrt (x ^ 2-a ^ 2)) (a \\ not \u003d 0)	F (x) \u003d \\ FRAC (1) (2a) L n \\ LVERT \\ FRAC (X-A) (X + A) \\ RVERT + C	\\ int \\ FRAC (DX) (\\ SQRT (X ^ 2-A ^ 2)) \u003d \\ FRAC (1) (2a) L N \\ LVERT \\ FRAC (X-A) (X + A) \\ RVERT + C
f (x) \u003d \\ tg x	F (x) \u003d - L n \\ LVERT \\ COS X \\ RVERT + C	\\ int \\ tg x dx \u003d - L n \\ LVERT \\ COS X \\ RVERT + C
f (x) \u003d \\ ctg x	F (x) \u003d L n \\ LVERT \\ SIN X \\ RVERT + C	\\ int \\ CTG X DX \u003d L N \\ LVERT \\ SIN X \\ RVERT + C
f (x) \u003d \\ FRAC (1) (\\ sin x)	F (x) \u003d L n \\ LVERT \\ TG \\ FRAC (X) (2) \\ RVERT + C	\\ int \\ FRAC (DX) (\\ Sin x) \u003d L n \\ LVERT \\ TG \\ FRAC (X) (2) \\ RVERT + C
f (x) \u003d \\ FRAC (1) (\\ COS X)	F (x) \u003d L n \\ LVERT \\ TG (\\ FRAC (X) (2) + \\ FRAC (\\ PI) (4)) \\ RVERT + C	\\ int \\ FRAC (DX) (\\ COS X) \u003d L N \\ LVERT \\ TG (\\ FRAC (X) (2) + \\ FRAC (\\ PI) (4)) \\ RVERT + C

Formula Newton Labitsa

Let be f (x) This feature, F. Her arbitrary primitive.

\\ int_ (a) ^ (b) f (x) dx \u003d f (x) | _ (a) ^ (b)\u003d F (B) - F (a)

where F (x) - Pred-like for F (X)

That is, the integral function f (X) The interval is equal to the difference in the sights at points b. and a..

Square of curvilinear trapezium

Curvilinear trapezium called a figure limited by a non-negative and continuous schedule on a segment of the function f., Ox axis and straight x \u003d A. and x \u003d B..

The area of \u200b\u200bthe curvilinear trapezium is found according to Newton Labitsa formula:

S \u003d \\ int_ (a) ^ (b) f (x) dx

Definition of primitive.

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.

Definition of an undefined integral.

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

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

The action of finding an unknown function according to its defined 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.

Based on the properties of the derivative, you can formulate and prove properties of an uncertain integral (Proph-shaped properties).

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.

Consider an example.

Example.

Find a primitive function whose value is united at x \u003d 1.

Decision.

We know from differential calculus that (It is enough to look into the table derivatives of the main elementary functions). In this way, . According to the second property . That is, we have many primitive. At x \u003d 1, we get a value. By condition, this value should be equal to one, therefore, C \u003d 1. The desired primitive will take a look.

Example.

Find an indefinite integral And the result check the differentiation.

Decision.

According to the sine formula of a double angle from trigonometry , so

Primed function and indefinite integral

Fact 1. Integration - action, inverse differentiation, namely, restoring the function according to a known derivative of this function. Function restored F.(x.) Called predo-shaped For function f.(x.).

Definition 1. Function F.(x. f.(x.) at some interval X.if for all values x. equality is performed from this gap F. "(x.)=f.(x.), that is, this feature f.(x.) is a derivative of a primitive function F.(x.). .

For example, a function F.(x.) \u003d SIN. x. is a primary for function f.(x.) \u003d COS. x. on the whole numerical straight, since with any value of the IKSA (sin. x.) "\u003d (COS x.) .

Definition 2. Uncertainly integral function f.(x.) It is called the totality of all its primitive. This uses recording

∫

f.(x.)dX.

where sign ∫ called the integral sign, function f.(x.) - a replacement function, and f.(x.)dX. - A concrete expression.

Thus, if F.(x.) - some kind of primary for f.(x.), T.

∫

f.(x.)dX. = F.(x.) +C.

where C. - arbitrary constant (constant).

To understand the meaning of many primitive functions as an indefinite integral, the following analogy is appropriate. Let there be a door (traditional wooden door). Its function is "to be a door." And what is the door made from? From wood. Therefore, a multitude of primitive integrated function "Be the door", that is, it is an indefinite integral, is the function "Being + C", where C is a constant, which in this context may indicate, for example, a tree of wood. Just as the door is made of wood using some tools, the derivative of the "made" function from the primitive function with the formulas that we learned by studying the derivative .

Then the table of the functions of common objects and the corresponding primitive ("to be the door" - "be tree", "be a spoon" - "be metal", etc.) is similar to the table of the main indefinite integrals, which will be shown slightly below. The table of uncertain integrals lists common functions with the indication of the primordial, of which these functions are made. In terms of the tasks to find a indefinite integral, such integrants are given, which without particular gravity can be integrated directly, that is, on the table of uncertain integrals. In the tasks, it is necessary to pre-convert to the tasks to preform so that you can use table integrals.

Fact 2. Restoring the function as a primitive, we must take into account an arbitrary constant (constant) C., so as not to write a list of primitive with different constants from 1 to infinity, you need to record many of the primitive with an arbitrary constant C.For example, as follows: 5 x.³ + p. So, an arbitrary constant (constant) enters the expression of primitive, since the primitive can be a function, for example, 5 x.³ + 4 or 5 x.³ + 3 and with differentiation 4 or 3, or any other constant is applied to zero.

We will put the integration task: for this function f.(x.) find such a function F.(x.), derivative of which equal f.(x.).

Example 1.Find a variety of features

Decision. For this feature, the function is function

Function F.(x.) called primitive for function f.(x.) if derivative F.(x.) Equal f.(x.), or that the same, differential F.(x.) Raven f.(x.) dX..

(2)

Consequently, the function is primitive for a function. However, it is not the only primary for. They also serve as functions

where FROM - Arbitrary constant. This can be seen differentiation.

Thus, if there is one first primary for the function, then it has an infinite multitude of primitive, differing in permanent term. All the primary functions are written in the above form. This follows from the following theorem.

Theorem (formal statement of fact 2).If a F.(x.) - Valid for function f.(x.) at some interval H., then any other primitive for f.(x.) At the same gap can be presented in the form F.(x.) + C.where FROM- Arbitrary constant.

In the following example, we already appeal to the integral table, which will be given in paragraph 3, after the properties of an indefinite integral. We do it before familiarization with the entire table, so that the essence of the foregoing is understood. And after the table and properties we will use them when integrating in all fullness.

Example 2.Find multiple features:

Decision. We find the sets of primitive functions, of which "these functions are made". When mentioning the formulas from the integral table, simply accept that there are such formulas, and we will study the table of uncertain integrals to be completely further.

1) applying formula (7) from the integral table with n. \u003d 3, we get

2) using formula (10) from the integral table with n. \u003d 1/3, we have

3) as

then by formula (7) when n. \u003d -1/4 Find

Under the sign of the integral write not the function itself f. , and her work on differential dX. . This is done primarily in order to indicate which variable is looking for a primitive. For example,

, ;

here, in both cases, the integrand function is equal, but its indefinite integrals in the considered cases are different. In the first case, this feature is considered as a function from a variable x. , and in the second - as a function from z. .

The process of finding an indefinable integral function is called integrating this function.

Geometric meaning of an indefinite integral

Let it be required to find a curve y \u003d f (x) And we already know that the tangent of the tilt angle at each of its point is the specified function f (x) The abscissions of this point.

According to the geometric meaning of the derivative, tangent tilt angle at this point of the curve y \u003d f (x) equal to the value of the derivative F "(x). So you need to find such a function F (x), for which F "(x) \u003d f (x). Function required in the task F (x) is a primary one f (x). The condition of the problem satisfies not one curve, but the family of curves. y \u003d f (x) - one of such curves, and every other curve can be obtained from her parallel transfer along the axis Oy..

Let's call a graph of a primitive function from f (x) integral curve. If a F "(x) \u003d f (x)then the graph of the function y \u003d f (x) There is an integral curve.

Fact 3. An uncertain integral is geometrically represented by the seven of all integrated curves As in the figure below. The remoteness of each curve from the start of coordinates is determined by an arbitrary constant (constant) integration C..

Properties of an indefinite integral

Fact 4. Theorem 1. The derivative of an indefinite integral is equal to the integrand function, and its differential is a source expression.

Fact 5. Theorem 2. Unexposed integral from differential function f.(x.) Equal function f.(x.) with an accuracy of a permanent term .

(3)

Theorems 1 and 2 show that differentiation and integration are mutually reverse operations.

Fact 6. Theorem 3. A constant multiplier in the integrand can be made for a sign of an indefinite integral .

Definition. The function f (x) is called primitive for the function f (x) at a given gap, if for any x from this gap "(x) \u003d f (x).

The main property of the primordial.

If f (x) is a primitive function f (x), then the function f (x) + C, where C is a perfect constant, it is also a primitive function f (x) (i.e., all the primitive functions f (x) are recorded in the form f (x) + s).

Geometric interpretation.

The graphs of all the primary functions of the F (X) are obtained from the graph of any one primitive parallel transfers along the OU axis.

PRINTING TABLE.

Rules of finding primary .

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

1. f ( X.) ± G ( X.) - Pred-like for F.( X.) ± G.( X.);

2. but F ( X.) - Pred-like for but F.( X.);

3. - Pervious for but F.( KX +. B.).

Tasks and tests on the topic "Pred-like"

PRINTING
Lessons: 1 tasks: 11 tests: 1
Derivative and primitive - Preparation for the exam in mathematics ege on mathematics
Tasks: 3.
Integral - Pred-like and integral grade 11
Lessons: 4 tasks: 13 tests: 1
Calculating areas with the help of integrals - Pred-like and integral grade 11
Lessons: 1 tasks: 10 tests: 1

Having studied this topic, you need to know what is called a primitive, its main property, geometric interpretation, the rules of finding primitive; To be able to find all the primitive functions using the table and the rules for finding primitive, as well as the primitive, passing through the specified point. Consider solving problems on this topic on the examples. Pay attention to the decisions.

Examples.

1. Find out whether the function f ( x.) = h. 3 – 3h. + 1 primary for function f.(x.) = 3(h. 2 – 1).

Decision: F "( x.) = (h. 3 – 3h. + 1) '\u003d 3 h. 2 – 3 = 3(h. 2 – 1) = f.(x.), i.e. F "( x.) = f.(x.), therefore, F (X) is a primitive for the function F (X).

2. Find all the primitive functions f (x):

but) f.(x.) = h. 4 + 3h. 2 + 5

Decision: Using the table and the rules of finding primitive, we get:

Answer:

b) f.(x.) \u003d sin (3 x. – 2)

Decision:

Perfect. Beautiful word.) To start a little Russian. This word is pronounced this way, and not "Pred-like" how it may seem. Pred-like is the basic concept of all integral calculus. Any integrals - indefinite, defined (with them you will get acquainted already in this semester), as well as double, triple, curvilinear, superficial (and these are the main heroes of the second course) - are built on this key concept. It has a complete sense to master. Go.)

Before getting to get acquainted with the concept of primitive, let's remember the most ordinary derivative. Without deepening into a boring theory of limits, increments of argument and other things, we can say that the derivative is found (or differentiation) Is just a mathematical operation function. And that's it. Any function is taken (let's say f (x) \u003d x 2) I. according to a certain rulesconverted by turning into new feature. And this is the most new feature and called derivative.

In our case, before the differentiation there was a function f (x) \u003d x 2, and after differentiation it became already another function f '(x) \u003d 2x.

Derivative - Because our new feature f '(x) \u003d 2x occurred from function f (x) \u003d x 2. As a result of the differentiation operation. And and with it, and not from some other function ( x 3., eg).

Roughly speaking, f (x) \u003d x 2 - this is mom, and f '(x) \u003d 2x - Her beloved daughter.) It is understandable. Go ahead.

Mathematics - the people are restless. For each action, they seek to find opposition. :) There is addition - there is a subtraction. There is multiplication - there is a division. Establishment - Extraction of the root. Sinus - Arksinus. Similarly, there is differentiation- So, there is and ... integration.)

And now we will put such an interesting task. We have, let's say, such a simple function f (x) \u003d 1. And we need to answer such a question:

Derived what function gives us a functionf.(x.) = 1?

In other words, seeing daughter, with the help of DNA analysis, calculate, who is her milf. :) So from what source Functions (let's call it f (x)) OUR occurred derivative Function f (x) \u003d 1? Or, in mathematical form, for what Functions f (x) Equality is performed:

F '(x) \u003d f (x) \u003d 1?

An example is elementary. I tried.) We simply select the function f (x) so that the equality worked. :) Well, how, picked up? Yes of course! F (x) \u003d x. Because:

F '(x) \u003d x' \u003d 1 \u003d f (x).

Of course, the mammy found F (x) \u003d x We must somehow call, yes.) Meet!

Perfect for functionf.(x.) This feature is calledF.(x.), the derivative of which is equalf.(x.), i.e. for which equality is rightF.’(x.) = f.(x.).

That's all. More scientific tricks. In strict definition, an additional phrase is added "At the interval". But so far we will not delve into these subtleties, because our primary task is to learn how to find these very primitive.

In our case, it turns out that the function F (x) \u003d x is an predo-shaped For function f (x) \u003d 1.

Why? Because F '(x) \u003d f (x) \u003d 1. The derivative of the ICA is a unit. No objections.)

The term "primitive" along the philistine means "Rhodonachable", "Parent", "ancestor". Immediately remember the native and loved one.) And the search itself is a primitive - this is the restoration of the original function according to its known derivative. In other words, this is an action, inverse differentiation. And that's all! This fascinating process itself is also called quite scientific integration. But by integrals - later. Patience, friends!)

Remember:

Integration is a mathematical operation of a function (as well as differentiation).

Integration - Operation, Inverse Differentiation.

Pred-like - the result of integration.

And now complicate the task. We will now find a primitive for function f (x) \u003d x. That is, we will find such a feature F (x) to its derivative I would be equal to ICSU:

F '(x) \u003d x

Who is friends with derivatives, perhaps something like something like:

(x 2) '\u003d 2x.

Well, respect and respect to those who remember the table of derivatives!) True. But there is one problem. Our initial function f (x) \u003d x, but (x 2) '\u003d 2 x.. Two X. And we have after differentiation should turn out just X.. Not okay. But…

We are a scientist with you. Certificates received.) And we know from school that both parts of any equality can be multiplied and divided into one and the same number (except for zero, of course)! So this arranged. So we realize this opportunity for yourself for good.)

After all, we want to right to remain clean X, right? And the deuce interferes ... here and take the ratio for the derivative (x 2) '\u003d 2x and divide both of its parts on this twice:

So, already clearing something. Go ahead. We know that any constant can take out a derivative sign.Like this:

All formulas in mathematics work both from left to right and on the contrary - right to left. This means that, with the same success, any constant can and Make under the sign of the derivative:

In our case, hiding a two in the denominator (or that the same, the coefficient 1/2) under the sign of the derivative:

And now carefully We look at our record. What do we see? We see equality that says that derivative something (this is something - In brackets) equals ICSU.

The resulting equality means that the desired primitive for the function f (x) \u003d x Serves function F (x) \u003d x 2/2 . That that stands in brackets under the touche. Directly in the sense of primitive.) Well, check the result. Find a derivative:

Excellent! The initial function was obtained f (x) \u003d x. From what was danced, to that and returned. This means that our primitive found right.)

What if f (x) \u003d x 2? What is its primitive? No problem! We know with you (again, from the rules of differentiation) that:

3x 2 \u003d (x 3) '

AND, that is,

Caught? Now we, imperceptibly for themselves, learned to consider first for any power function f (x) \u003d x n. In the mind.) Take the source n., increase it per unit, and in the quality of compensation we divide the entire design on n + 1.:

The resulting formula, by the way, is valid not only for a natural figure degree n.But for any other - negative, fractional. This makes it easy to find primitive from simple frains and roots.

For example:

Naturally, n ≠ -1. Otherwise, in the denominator of the formula, it turns out zero, and the formula loses its meaning.) About this particular case n \u003d -1. a little bit later.)

What is an uncertain integral? Table integrals.

Let's say what is equal to the function F (x) \u003d x? Well, the unit, one - I hear dissatisfied answers ... everything is true. Unit. But ... for function G (x) \u003d x + 1 derivative will also be equal to one:

Also, the derivative will be equal to one and for the function x + 1234. , and for function x-10. and for any other type of type x + C. where FROM - any constant. For the derivative of any constant is zero, and from the addition / subtraction of zero, none is cold or hot.)

It turns out ambiguity. It turns out that for the function f (x) \u003d 1 Pred-like serve not only a function F (x) \u003d x , but also a function F 1 (x) \u003d x + 1234 and function F 2 (x) \u003d X-10 etc!

Yes. That's the way.) U anyone ( continuous on the interval) Functions There is no one very primitive, but infinitely a lot - Whole family! Not one mother or dad, but a whole pedigree, yeah.)

But! All our relatives-face combines one important property. That they are relatives.) The property is so important that in the process of the analysis of the techniques of integration, we have repeatedly remember. And we will remember for a long time.)

Here it is, this property:

Any two primitive F. 1 (x.) I.F. 2 (x.) From the same functionf.(x.) differ in the constant:

F. 1 (x.) - F. 2 (x.) \u003d S.

Who is interested in proof - Stater the literature or abstract lectures.) Okay, so be, I will prove. The benefit of the proof here is elementary, in one action. Take equality

F. 1 (x.) - F. 2 (x.) \u003d S.

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

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

What does this property say? And that two different primary from the same function f (x) can't be different some expression with X . Only strictly to the constant! In other words, if we have some kind of schedule one of the primordial (Let it be f (x)), then graphics all others Our primitive are built by parallel transfer of the Graph F (X) along the axis of the game.

Let's see how it looks like an example f (x) \u003d x. All its primary, as we already know, have a general view. F (x) \u003d x 2/2 + c . In the picture it looks like infinite many parabolobtained from the "main" parabola y \u003d x 2/2 shift along the axis oy up or down depending on the value of the constant FROM.

Remember the school building function y \u003d f (x) + a Shift graphics y \u003d f (x) On "a" units along the axis of the game?) So here is the same.)

And pay attention: our parabolas nowhere are not intersecting!It is natural. After all, two different functions y 1 (x) and y 2 (x) will inevitably correspond to two different values \u200b\u200bof the constant – With 1. and With 2.

Therefore, the equation y 1 (x) \u003d y 2 (x) never has solutions:

C 1 \u003d C 2

x ε ∅ , as C 1 ≠ C2

And now we smoothly approach the second cornerstone concept of integral calculus. As we have just installed, in any function f (x) there is an infinite set of primitive F (x) + C, differing from each other to the constant. This is the most infinite set also has its own special name.) Well, I ask you to love and complain!

What is an uncertain integral?

Many of all primitive for function f.(x.) Called uncertain integralfrom functionf.(x.).

That's all definition.)

"Uncertain" - Because the set of all primitive for the same function infinitely. Too many different options.)

"Integral" - with a detailed decoding of this brutal word, we will get acquainted in the next big section dedicated to defined integrals. In the meantime, in coarse form, we will consider something by integral general, single, whole. And integration - an association, generalizationIn this case, the transition from the private (derivative) to general (primitive). Something like that.

Denotes an indefinite integral like this:

Read the same way as written: integral EF from X DE X. Or integral from EF from X DE X.Well, you understood.)

Now we will deal with the notation.

∫ - integral icon. The point is the same as the barcode for the derivative.)

d. - icondifferential. Not afraid! Why it is needed there - just below.

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

f (x) dx - inhibitory expression. Or, roughly speaking, "filling" integral.

According to the meaning of an indefinite integral,

Here F (x) - that SAMIA pRINTING For function f (x)that we are somehow found themselves.How exactly found - do not essence. For example, we found that F (x) \u003d x 2/2 for f (x) \u003d x.

"FROM" - arbitrary constant. Or, more scientifically, integral Constanta. Or integration constant. All one.)

And now let us return to our very first examples on the search for a primitive. In terms of an indefinite integral, you can now boldly 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 entire infinite set of primitive highlights the line, which passes through a specified point.

What's the point. From the initial infinite set of primitive (i.e. indefinite integral) It is necessary to highlight the curve that will pass through the specified point. With somehow specific coordinates.Such a task is always and everywhere occurring at an initial acquaintance with integrals. Both at school and in the university.

Typical problem:

Among the sets of all the primitive functions f \u003d x, select the one that passes through the point (2; 2).

We begin to think your head ... Many of all first-handed - it means, you must first integrate our original function.That is, X (x). With this we were engaged slightly higher and got such an answer:

And now we understand what we got. We got not one function, but whole family of functions. Which ones? View y \u003d x 2/2 + C . Conducting the value of the Constant C. And this is the meaning of the constant to us and now have to "catch".) Well, what do you catch?)

Our fishing rod - family of Curves (Parabola) y \u003d x 2/2 + c.

Constants - these are fishery. Many-many. But each there is a hook and bait.)

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

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

y (2) \u003d 2

From here is easily searched C \u003d 0..

What does this mean? This means that from the entire infinite set parabola typey \u003d x 2/2 + Conly parabola with constant C \u003d 0 It is suitable for us! Namely:y \u003d x 2/2. And only she. Only this parabola will pass through the point you need (-2; 2). A B.ses other parabolas from our family pass through this point no longer be.Through some other points of the plane - yes, but through the point (2; 2) - no longer. Caught?

For clarity here are two pictures - all the parabola family (ie, an indefinite integral) and some kind concrete Parabolacorresponding specific value of the constant and passing through specific point:

See how important it is to take into account the constant FROM When integrating! So we do not neglect this beak "C" and do not forget to attribute to the final answer.

And now we'll figure it out, why inside the integrals everywhere the symbol hangs out dX. . Students forget about him often ... And this is, by the way, also a mistake! And rather rude. The thing is that integration is an operation, inverse differentiation. And what exactly is differentiation results? Derivative? True, but not quite. Differential!

In our case, for function f (x) Differential its primary F (x), will be:

To whom this chain is incomprehensible - urgently repeat the definition and meaning of the differential and how it is revealed! Otherwise, in the integrals you will slow down mercilessly ....

Let me remind you in the coarse philistine form that the differential of any function f (x) is just a work f '(x) dx. And that's all! Take a derivative and multiply it on the differential argument (i.e. DX). That is, any differential, in fact, comes down to the calculation of the usual derivative.

Therefore, strictly speaking, the integral "takes" not from functions f (x)as it is considered, and from differential f (X) DX! But, in the simplified version, it is customary to say that "The integral is taken from the function". Or: "The function f integrates(x)". This is the same. And we will talk in the same way. But about the icon dX. At the same time, you will not forget! :)

And now I will tell you how to not forget it when recording. Imagine first that you calculate the usual derivative of the ICS variable. How do you usually write it?

So: f '(x), y' (x), y 'x. Or more solid, through the ratio of differentials: DY / DX. All these records show us that the derivative is taken on ICSU. And not by "Igrek", "TE" or some other variable there.)

Also in integrals. Record ∫ F (x) DX U.S. too as if Indicates that integration is carried out precisely by variable ix. Of course, it's all very simplistic and rude, but it is clear, I hope. And chances forget attribute omnipresent dX. sharply decline.)

So, that the same uncertain integral - dealt with. Perfectly.) Now it would be nice to learn these most indefinite integrals calculate. Or, simply speaking, "take". :) And here the students are waiting for two news - good and not very. So far, we start with good.)

The news is good. For integrals, as well as for derivatives, there is its own plate. And all the integrals that we will meet on the way, even the most terrible and trusted, we according to a certain rules We will somehow reduce the most tabular.)

So, here she is table integrals!

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

The third group of formulas (trigonometry), as can be guessed, is obtained by simply appealing the corresponding formulas for derivatives.

For example:

With the fourth group of formulas (indicative function) - everything is similar.

But the four latest groups of formulas (5-8) for us new. How did they come from and for what such merits these are precisely these exotic functions, suddenly, entered the table of the main integrals? What do these groups of functions are allocated against the background of other functions?

So developed historically in the development process integration methods . When we train to take the most and most diverse integrals, you will understand that the integrals from the functions listed in the table are very and very often. It is often so often that mathematics attributed them to tabular.) Through them are very many other integrals, from more complex structures.

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

Everything is fine. Did not deceive mathematics. :)

Table of integrals, as well as a table of derivatives, it is desirable to know by heart. In any case, the first four groups of formulas. It is not as hard as it seems at first glance. Understand by heart the last four groups (with fractions and roots) until Do not. Anyway, at first, you will be confused where the logarithm is to write, where Arcthangenes, where Arksinus, where 1 / A, where 1 / 2a ... The exit here is one - to solve more examples. Then the table itself will gradually and remember, and the doubts nibble will stop.)

Particularly inquisitive faces, looking at the table, can ask: and where in the table integrals from other elementary "school" functions - tangent, logarithm, "arches"? Let's say why the table has an integral from sinus, but there is no, let's say, the integral from Tangent tG X.? Or not there integral from logarithm lN X.? From Arksinus arcsin X.? What are they worse? But it is full of "left" functions - with roots, fractions, squares ...

Answer. No worse.) Just the above-mentioned integrals (from tangent, logarithm, arxinus, etc.) are not tabular . And they are in practice much less frequently than those are presented in the table. Therefore, know by heartWhat they are equal, not necessarily. Just know enough like they calculate.)

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

How will you memorize? :) Will you not? And do not.) But do not worry, we will definitely find all such integrals. In the appropriate lessons. :)

Well, now go to the properties of an indefinite integral. Yes, yes, nothing to do! A new concept is introduced - immediately and some of its properties are considered.

Properties of an indefinite integral.

Now not very good news.

In contrast to differentiation, general Standard Integration RulesFair for all occasions, in mathematics there is no. It is fantastic!

For example, you know everything perfectly (I hope!) That anyone composition any Two functions f (x) · g (x) is differentiated as follows:

(f (x) · g (x)) '\u003d f' (x) · g (x) + f (x) · g '(x).

Anyone Private differentiates like this:

And any complex function, whatever over with it, is differentiated as follows:

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 pass: for the work, private (fraction), as well as the complex function of general integration formulas does not exist! There is no standard rules! Rather, they are. It is in vain mathematics offended.) But, first, they are much smaller than the general rules for differentiation. And secondly, most integration methods we will talk about in the following lessons, very, very specific. And are valid only for a certain, very limited class of functions. Let's just say for fractional rational functions. Or some more.

And some integrals, although exist in nature, but at all are not expressed in any way through the elementary "school" functions! Yes, and such integrals are full! :)

That is why integration is a much more time-consuming and painstaking lesson than differentiation. But there is also its own highlight. The occupation is creative and very exciting.) And if you are well digest on the integral table and master at least two basic receptions, which we will talk about (and), then you will like the integration. :)

Now let's get acquainted, actually with the properties of an indefinite integral. They are all nothing. Here they are.

The first two properties are completely similar to the same properties for derivatives and are called. properties of the linearity of an indefinite integral . Everything is simple and logical here: the integral from the amount / difference is equal to the amount / difference of the integrals, and the constant multiplier can be taken out of the integral sign.

And here is the following three properties for us fundamentally new. We will analyze them in more detail. They sound in Russian as follows.

Third Property

The derivative of the integral is equal to the integrand function

Everything is simple, as in a fairy tale. If you integrate the function, and then back to find a derivative of the result, then ... it turns out the initial integrand function. :) This property can always (and necessary) use to verify the final integration result. Calculated the integral - Differentiate the answer! Received a detailed function - approx. They did not get - it means somewhere they have accumulated. Look for an error.)

Of course, in the answer can be obtained so brutal and bulky functions, which is back to differentiate their reluctance, yes. But better, if possible, try to check ourselves. At least in those examples where it is easy.)

Fourth property

Differential from the integral is equal to the image .

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

Fifth property

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

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

These are the useful properties. I am not going to encourage with their strict evidence. Wishing to offer it yourself. Directly over the sense of derivative and differential. I will prove only the last, fifth property, for it is less obvious.

So, we have a statement:

I pull out the "filling" of our integral and disclose, according to the definition of differential:

Just in case, I remind you that, according to our designation derivative and primitive, F.’(x.) = f.(x.) .

Insert now our result back inside the integral:

Received exactly definition of an indefinite integral (Let me forgive me Russian)! :)

That's all.)

Well. This is our initial acquaintance with the mysterious world of integrals, I consider it. Today I propose to round. We are already armed enough to go into intelligence. If not a machine gun, then at least a water pistol basic properties and table. :) In the next lesson, we are already waiting for the simplest innocuous examples of integrals on the direct application of the table and written properties.

See you!