The second order partial derivative has the form. Partial derivatives of the second order of a function of three variables

Let a function be given. Since x and y are independent variables, one of them can change, while the other retains its value. Give the independent variable x an increment while keeping the y value unchanged. Then z will receive an increment, which is called the partial increment of z in x and is denoted. So, .

Similarly, we obtain the partial increment of z with respect to y:.

The total increment of the function z is determined by equality.

If there is a limit, then it is called the partial derivative of the function at a point with respect to the variable x and is denoted by one of the symbols:

.

Partial derivatives with respect to x at a point are usually denoted by symbols .

The partial derivative of the variable y is defined and denoted similarly:

Thus, the partial derivative of a function of several (two, three or more) variables is defined as the derivative of a function of one of these variables, provided that the values \u200b\u200bof the remaining independent variables are constant. Therefore, the partial derivatives of a function are found by the formulas and rules for calculating the derivatives of a function of one variable (in this case, respectively, x or y are considered constant).

Partial derivatives are called first order partial derivatives. They can be viewed as functions of. These functions can have partial derivatives, which are called second order partial derivatives. They are defined and labeled as follows:

; ;

; .

Differentials 1 and 2 of the order of a function of two variables.

The total differential of a function (formula 2.5) is called a first-order differential.

The formula for calculating the total differential is as follows:

(2.5) or where,

partial differentials of the function.

Let the function have continuous second order partial derivatives. The differential of the second order is determined by the formula. Let's find it:

Hence: ... This is symbolically written as follows:

.

UNDEFINED INTEGRAL.

Antiderivative of a function, indefinite integral, properties.

The function F (x) is called antiderivativefor a given function f (x), if F "(x) \u003d f (x), or, which is the same, if dF (x) \u003d f (x) dx.

Theorem. If the function f (x), defined in some interval (X) of finite or infinite length, has one antiderivative, F (x), then it also has infinitely many antiderivatives; they are all contained in the expression F (x) + C, where C is an arbitrary constant.

The collection of all antiderivatives for a given function f (x), defined in some interval or on some segment of finite or infinite length, is called indefinite integral from the function f (x) [or from the expression f (x) dx] and is denoted by the symbol.

If F (x) is one of the antiderivatives for f (x), then according to the antiderivatives theorem

, where C is an arbitrary constant.

By the definition of the antiderivative F "(x) \u003d f (x) and, therefore, dF (x) \u003d f (x) dx. In formula (7.1), f (x) is called the integrand, and f (x) dx is the integrand. expression.

Partial derivatives of functions of several variables are functions of the same variables. These functions, in turn, can have partial derivatives, which we will call the second partial derivatives (or second order partial derivatives) of the original function.

So, for example, a function of two variables has four second-order partial derivatives, which are defined and denoted as follows:

The three-variable function has nine second-order partial derivatives:

The partial derivatives of the third and higher order of a function of several variables are defined and denoted in a similar way: the partial derivative of the order of a function of several variables is the partial derivative of the first order of the partial derivative of the order of the same function.

For example, the third-order partial derivative of a function is the first-order partial derivative in y of the second-order partial derivative

A partial derivative of the second or higher order, taken over several different variables, is called a mixed partial derivative.

For example, partial derivatives

are mixed partial derivatives of a function of two variables.

Example. Find second order mixed partial derivatives of a function

Decision. Find the partial derivatives of the first order

Then we find the mixed partial derivatives of the second order

We see that the mixed partial derivatives and differing from each other only in the order of differentiation, that is, in the sequence in which differentiation is made with respect to different variables, turned out to be identically equal. This result is not accidental. The following theorem holds for mixed partial derivatives, which we accept without proof.

We continue the favorite topic of mathematical analysis - derivatives. In this article we will learn how to find partial derivatives of a function of three variables: first derivatives and second derivatives. What do you need to know and be able to do to master the material? Believe it or not, first of all, you need to be able to find the "ordinary" derivatives of a function of one variable - at a high or at least average level. If they are very tight, then start with a lesson How to find a derivative? Secondly, it is very important to read the article and comprehend-solve, if not all, then most of the examples. If this has already been done, then walk with me with a confident gait, it will be interesting, even enjoy it!

Methods and principles of finding partial derivatives of a function of three variables are actually very similar to partial derivatives of functions of two variables. As a reminder, the function of two variables has the form, where “x” and “game” are independent variables. Geometrically, a function of two variables is a surface in our three-dimensional space.

The function of three variables has the form, while the variables are called independent variables or arguments, the variable is called dependent variable or function... For example: - function of three variables

And now a little about science fiction films and aliens. You can often hear about 4D, 5D, 10D, etc. spaces. Nonsense or not?
After all, a function of three variables implies the fact that all things happen in four-dimensional space (indeed, there are four variables). The graph of a function of three variables is the so-called hypersurface... It is impossible to imagine it, since we live in three-dimensional space (length / width / height). So that you don't get bored with me, I propose a quiz. I will ask a few questions, and those who wish can try to answer them:

- Is there a fourth, fifth, etc. in the world? measurements in the sense of a common understanding of space (length / width / height)?

- Is it possible to build four-dimensional, five-dimensional, etc. space in the broadest sense of the word? That is, to give an example of such a space in our life.

- Is it possible to travel to the past?

- Is it possible to travel to the future?

- Do aliens exist?

You can choose one of four answers to any question:
Yes / No (science is prohibited) / Science is not prohibited / I don't know

Whoever answers all the questions correctly, most likely has some thing ;-)

I will gradually give answers to questions during the lesson, do not miss examples!

Actually, they flew. And immediately the good news: for a function of three variables, the rules of differentiation and the table of derivatives are valid... This is why you need to be good at handling "ordinary" derivatives of functions one variable. There are very few differences!

Example 1

Decision:It is not hard to guess - for a function of three variables there are three partial derivatives of the first order, which are denoted as follows:

Or - a partial derivative with respect to "x";
or - the partial derivative with respect to "y";
or - partial derivative with respect to "z".

The designation with a stroke is more common, but the compilers of collections, manuals in the conditions of problems are very fond of using just bulky designations - so do not get lost! Perhaps not everyone knows how to correctly read these "scary fractions" aloud. Example: it should be read as follows: "de u po de x".

Let's start with the x-derivative:. When we find the partial derivative with respect to , then the variables and are considered constants (constant numbers). And the derivative of any constant, oh, grace, is zero:

Pay attention to the subscript right away - no one forbids you to mark that they are constants. It is even more convenient, for beginners I recommend using just such a recording, there is less risk of confusion.

(1) We use the linearity properties of the derivative, in particular, we move all constants outside the sign of the derivative. Please note that in the second term, the constant does not need to be taken out: since the "game" is a constant, then it is also a constant. In the term, the "usual" constant 8 and the constant "z" are taken out beyond the sign of the derivative.

(2) Find the simplest derivatives without forgetting that are constants. Next, we comb the answer.

Partial derivative. When we find the partial derivative with respect to "game", then the variables and are considered constants:

(1) We use the linearity properties. And again, note that the terms are constants, which means that nothing needs to be taken out beyond the sign of the derivative.

(2) Find derivatives, not forgetting that constants. Let's simplify the answer further.

And finally, the partial derivative. When we find the partial derivative with respect to "z", then the variables and are considered constants:

General rule obvious and unpretentious: When we find the partial derivativefor any independent variable, thenthe other two independent variables are considered constants.

When preparing these tasks, you should be extremely careful, in particular, you must not lose subscripts (which indicate which variable is being differentiated). Loss of the index will be a BADNESS. Hmmm…. It's funny if, after such intimidation, I myself miss them somewhere)

Example 2

Find the partial derivatives of the first order of a function of three variables

This is an example for a do-it-yourself solution. Complete solution and answer at the end of the tutorial.

The two examples considered are quite simple and, having solved several similar problems, even the teapot will get used to dealing with them orally.

To unload, let's return to the first question of the quiz: Is there a fourth, fifth, etc. in the world. measurements in the sense of a common understanding of space (length / width / height)?

Correct answer: Science is not forbidden... All fundamental mathematical axioms, theorems, mathematical apparatus are excellent and consistently work in a space of any dimension. It is possible that somewhere in the Universe there are hypersurfaces beyond the control of our mind, for example, a four-dimensional hypersurface, which is given by a function of three variables. Or maybe there are hypersurfaces next to us, or even we are right there, just our vision, other senses, consciousness are capable of perceiving and comprehending only three dimensions.

Let's go back to the examples. Yes, if someone is heavily loaded with the quiz, it is better to read the answers to the following questions after you learn how to find the partial derivatives of a function of three variables, otherwise I will take out my whole brain in the course of the article \u003d)

In addition to the simplest Examples 1 and 2, in practice there are tasks that can be called a small puzzle. Such examples, to my chagrin, fell out of sight when I created the lesson Partial derivatives of a function of two variables... Making up for lost time:

Example 3

Decision: It seems that everything is simple here, but the first impression is deceiving. When finding partial derivatives, many will guess on the coffee grounds and make mistakes.

Let's analyze the example consistently, clearly and clearly.

Let's start with the partial derivative with respect to "x". When we find the partial derivative with respect to "x", then the variables are considered constants. Therefore, the exponent of our function is also a constant. For dummies, I recommend the following solution: on a draft, change the constant to a specific positive integer, for example, to "five". The result will be a function of one variable:
or you can also write it like this:

it sedate a function with a compound radix (sine). By :

Now we remember that, in this way:

On a clean copy, of course, the solution should be formalized as follows:

We find the partial derivative with respect to the "game", are considered constants. If "x" is a constant, then it is also a constant. On the draft, we do the same trick: we replace, for example, with 3, "z" - we replace it with the same "five". The result is again a function of one variable:

it indicative function with a complex exponent. By the rule for differentiating a complex function:

Now we remember our replacement:

In this way:

On a clean copy, of course, the design should look nice:

And the mirror case with a partial derivative with respect to "z" (- constants):

With some experience, the analysis can be carried out mentally.

We carry out the second part of the task - we will compose the differential of the first order. It is very simple, by analogy with a function of two variables, the first-order differential is written by the formula:

In this case:

And business then. Note that in practical problems the full differential of the first order of functions of three variables is required to be made much less often than for a function of two variables.

A funny example for an independent solution:

Example 4

Find the partial derivatives of the first order of a function of three variables and compose the total differential of the first order

Complete solution and answer at the end of the tutorial. If you have any difficulties, use the considered "Dummy" algorithm, it is guaranteed to help. And another useful tip - do not hurry... Even I cannot quickly solve such examples.

We digress and analyze the second question: Is it possible to build a four-dimensional, five-dimensional, etc. space in the broadest sense of the word? That is, to give an example of such a space in our life.

Correct answer: Yes... Moreover, it is very easy. For example, we add the fourth dimension to the length / width / height - time. The popular four-dimensional space-time and the well-known theory of relativity, neatly stolen by Einstein from Lobachevsky, Poincaré, Lorentz and Minkowski. Also, not everyone knows. Why does Einstein get the Nobel Prize? There was a terrible scandal in the scientific world, and the Nobel Committee formulated the merit of the plagiarist approximately as follows: "For the general contribution to the development of physics." So that's it. Einstein's Troechnik brand is pure promotion and PR.

It is easy to add a fifth dimension to the considered four-dimensional space, for example: atmospheric pressure. And so on, so on, so on, how many dimensions you set in your model - so many will be. In the broadest sense of the word, we live in a multidimensional space.

Let's analyze a couple of typical tasks:

Example 5

Find the partial derivatives of the first order at a point

Decision: A task in this formulation is often encountered in practice and involves the following two actions:
- you need to find the partial derivatives of the first order;
- it is necessary to calculate the values \u200b\u200bof partial derivatives of the 1st order at a point.

We decide:

(1) We have a complex function, and at the first step we should take the derivative of the arctangent. In this case, we, in fact, calmly use the tabular formula for the derivative of the arctangent. By the rule for differentiating a complex functionthe result must be multiplied by the derivative of the internal function (embedding):.

(2) We use the linearity properties.

(3) And we take the remaining derivatives, not forgetting that they are constants.

According to the condition of the task, it is necessary to find the value of the found partial derivative at the point. Let's substitute the coordinates of the point into the found derivative:

The advantage of this task is the fact that other partial derivatives are found in a very similar pattern:

As you can see, the solution pattern is almost the same.

Let's calculate the value of the found partial derivative at the point:

And, finally, the z-derivative:

Done. The solution could be formulated in another way: first, find all three partial derivatives, and then calculate their values \u200b\u200bat a point. But, it seems to me, the given method is more convenient - they just found the partial derivative, and immediately, without leaving the checkout, calculated its value at the point.

It is interesting to note that geometrically a point is a very real point in our three-dimensional space. The values \u200b\u200bof the function and derivatives are already the fourth dimension, and nobody knows where it is geometrically. As they say, no one crawled around the Universe with a tape measure, did not check.

Once again the philosophical theme has gone, consider the third question: Is it possible to travel into the past?

Correct answer: No... Traveling into the past contradicts the second law of thermodynamics about the irreversibility of physical processes (entropy). So do not dive into the pool without water, please, the event can be turned back only in the video \u003d) Folk wisdom has not in vain come up with the opposite law of everyday life: "Measure seven times, cut once." Although, in fact, a sad thing, time is one-directional and irreversible, none of us will get younger tomorrow. And sci-fi films like The Terminator are scientifically nonsense. It is absurd also from the point of view of philosophy - when the Consequence, returning to the past, can destroy its own Cause. ...

It's more interesting with the derivative with respect to "z", although it's still almost the same:

(1) Move the constants outside the sign of the derivative.

(2) Here again, the product of two functions, each of which depends from the "live" variable "z". In principle, you can use the formula for the derivative of the quotient, but it is easier to go the other way - to find the derivative of the product.

(3) A derivative is a tabular derivative. The second term contains the familiar derivative of a complex function.

Example 9

Find the partial derivatives of the first order of a function of three variables

This is an example for a do-it-yourself solution. Think about how to rationally find this or that partial derivative. Complete solution and answer at the end of the tutorial.

Before moving on to the final examples of the lesson and consider partial derivatives of the second order functions of three variables, once again I will cheer everyone up with the fourth question:

Is travel to the future possible?

Correct answer: Science is not forbidden... Paradoxically, there is no mathematical, physical, chemical or other natural science law that would prohibit travel to the future! Sounds nonsense? But almost everyone in life had a premonition (moreover, not supported by any logical arguments) that this or that event would happen. And it happened! Where did the information come from? From the future? Thus, fantastic films about a journey into the future, and, by the way, predictions of all kinds of fortune-tellers, psychics cannot be called such nonsense. At least, science has not refuted this. Everything is possible! So, when I was in school, CDs and flat-panel monitors from films seemed incredible to me.

The well-known comedy "Ivan Vasilyevich Changes His Profession" is half fiction (as a maximum). No scientific law forbade Ivan the Terrible to be in the future, but it is impossible for two peppers to be in the past and fulfill the duties of a tsar.

Each partial derivative (by x and by y) of a function of two variables is the ordinary derivative of a function of one variable with a fixed value of the other variable:

(Where y\u003d const),

(Where x\u003d const).

Therefore, partial derivatives are calculated from formulas and rules for calculating the derivatives of functions of one variable , while considering the other variable constant (constant).

If you do not need an analysis of examples and the minimum of theory required for this, but only need a solution to your problem, then go to online partial derivatives calculator .

If it's hard to concentrate in order to keep track of where the constant is in the function, then you can substitute any number instead of a variable with a fixed value in the draft example of the example - then you can quickly calculate the partial derivative as an ordinary derivative of a function of one variable. You just need to remember to return the constant (variable with a fixed value) to its place during finishing.

The property of partial derivatives described above follows from the definition of a partial derivative, which can be found in exam questions. Therefore, to familiarize yourself with the definition below, you can open a theoretical reference.

Continuity of a function z= f(x, y) at a point is defined similarly to this concept for a function of one variable.

Function z = f(x, y) is called continuous at a point if

The difference (2) is called the total increment of the function z(it is obtained by incrementing both arguments).

Let the function z= f(x, y) and point

If function change zoccurs when only one of the arguments changes, for example x, with a fixed value of another argument y, then the function will be incremented

called the partial function increment f(x, y) by x.

Considering changing the function zdepending on the change in only one of the arguments, we actually go to the function of one variable.

If there is a finite limit

then it is called the partial derivative of the function f(x, y) by argument xand is indicated by one of the symbols

(4)

The partial increment is defined similarly zby y:

and the partial derivative f(x, y) by y:

(6)

Example 1.

Decision. We find the partial derivative with respect to the variable "x":

(yfixed);

Find the partial derivative with respect to the "game" variable:

(xfixed).

As you can see, it does not matter to what extent the variable that is fixed: in this case, it is just some number that is a factor (as in the case of the usual derivative) of the variable over which we find the partial derivative. If the fixed variable is not multiplied by the variable over which we find the partial derivative, then this lonely constant, it makes no difference to what extent, as in the case of the ordinary derivative, it vanishes.

Example 2.The function is given

Find partial derivatives

(by x) and (by game) and calculate their values \u200b\u200bat the point AND (1; 2).

Decision. With a fixed y the derivative of the first term is found as the derivative of the power function ( table of derivatives of functions of one variable):

.

With a fixed x the derivative of the first term is found as the derivative of the exponential function, and the second as the derivative of the constant:

Now we calculate the values \u200b\u200bof these partial derivatives at the point AND (1; 2):

You can check the solution of problems with partial derivatives on partial derivatives calculator online .

Example 3. Find Partial Derivatives of a Function

Decision. In one step we find

(y xas if the sine argument were 5 x: in the same way 5 appears before the function sign);

(x is fixed and in this case is a factor at y).

You can check the solution of problems with partial derivatives on partial derivatives calculator online .

Partial derivatives of functions of three or more variables are defined similarly.

If each set of values \u200b\u200b( x; y; ...; t) of independent variables from the set Dmatches one specific value uof the multitude Ethen ucalled the function of variables x, y, ..., tand denote u= f(x, y, ..., t).

There is no geometric interpretation for functions of three or more variables.

Partial derivatives of a function of several variables are also determined and calculated under the assumption that only one of the independent variables changes, while the others are fixed.

Example 4. Find Partial Derivatives of a Function

.

Decision. y and z fixed:

x and z fixed:

x and y fixed:

Find partial derivatives yourself and then see solutions

Example 5.

Example 6.Find the partial derivatives of the function.

The partial derivative of a function of several variables has the same the mechanical meaning is that the derivative of a function of one variable , is the rate of change of the function relative to the change in one of the arguments.

Example 8. Quantitative flow rate Prailway passengers can be expressed by the function

where P- the number of passengers, N- the number of residents of the corresponding points, R- distance between points.

Partial derivative of a function Pby Requal to

shows that the decrease in the flow of passengers is inversely proportional to the square of the distance between the corresponding points for the same number of inhabitants in the points.

Partial derivative Pby Nequal to

shows that the increase in the flow of passengers is proportional to the doubled number of inhabitants of settlements with the same distance between the settlements.

You can check the solution of problems with partial derivatives on partial derivatives calculator online .

Full differential

The product of a partial derivative and the increment of the corresponding independent variable is called a partial differential. Partial differentials are designated as follows:

The sum of the partial differentials over all independent variables gives the total differential. For a function of two independent variables, the total differential is expressed by the equality

(7)

Example 9.Find the Total Differential of a Function

Decision. The result of using formula (7):

A function that has a total differential at each point of a certain region is called differentiable in this region.

Find the full differential yourself and then see the solution

As in the case of a function of one variable, the differentiability of a function in a certain region implies its continuity in this region, but not vice versa.

Let us formulate without proofs a sufficient condition for the differentiability of a function.

Theorem.If the function z= f(x, y) has continuous partial derivatives

in a given area, then it is differentiable in this area and its differential is expressed by formula (7).

It can be shown that, just as in the case of a function of one variable, the differential of the function is the main linear part of the increment of the function, so in the case of a function of several variables, the total differential is the main, linear with respect to the increments of independent variables, part of the total increment of the function.

For a function of two variables, the total increment of the function has the form

(8)

where α and β are infinitesimal for and.

Partial derivatives of higher orders

Partial derivatives and functions f(x, y) are themselves some functions of the same variables and, in turn, can have derivatives with respect to different variables, which are called higher-order partial derivatives.

The general principle of finding second-order partial derivatives of a function of three variables is similar to the principle of finding second-order partial derivatives of a function of two variables.

In order to find the partial derivatives of the second order, you must first find the partial derivatives of the first order or, in another notation:

There are nine partial derivatives of the second order.

The first group is the second derivatives with respect to the same variables:

Or - the second derivative with respect to "x";

Or - the second derivative with respect to "y";

Or - the second derivative with respect to "z".

The second group is mixedpartial derivatives of the 2nd order, there are six of them:

Or - mixedderivative "by x y";

Or - mixedy-x derivative;

Or - mixed derivative "on x z";

Or - mixedderivative "on zet x";

Or - mixedderivative "yrek zet";

Or - mixed derivative "by zet y".

As in the case of a function of two variables, when solving problems, one can focus on the following equalities of mixed derivatives of the second order:

Note: Strictly speaking, this is not always the case. For the mixed derivatives to be equal, the requirement of their continuity must be satisfied.

Just in case, a few examples of how to correctly read this disgrace out loud:

- “two strokes, twice in a game”;

- "de two u po de zet square";

- "have two strokes on x on z";

- "de two u po de zet po de yrek."

Example 10

Find all partial derivatives of the first and second order for a function of three variables:

.

Decision:First, we find the partial derivatives of the first order:

We take the found derivative

and differentiate it by "game":

We take the found derivative

and differentiate it by "x":

The equality is satisfied. Okay.

Let's deal with the second pair of mixed derivatives.

We take the found derivative

and differentiate it by "z":

We take the found derivative

and differentiate it by "x":

The equality is satisfied. Okay.

We deal with the third pair of mixed derivatives in a similar way:

The equality is satisfied. Okay.

After the work done, it can be guaranteed that, firstly, we correctly found all the partial derivatives of the first order, and secondly, we correctly found the mixed partial derivatives of the second order.

It remains to find three more partial derivatives of the second order, here, in order to avoid errors, you should focus as much as possible:

Done. Again, the task is not so much difficult as it is voluminous. The solution can be shortened and referred to the mixed partial differential equalities, but in this case there will be no verification. So you better take your time and find allderivatives (besides, it may be required by the teacher), or, as a last resort, check on a draft.

Example 11

Find all partial derivatives of the first and second order for a function of three variables

.

This is an example for a do-it-yourself solution.

Solutions and Answers:

Example 2:Decision:

Example 4:Decision: Let's find the partial derivatives of the first order.

Let's compose the total differential of the first order:

Example 6:Decision: M(1, -1, 0):

Example 7:Decision: Let us calculate the partial derivatives of the first order at the point M(1, 1, 1):

Example 9:Decision:

Example 11:Decision: Let's find the partial derivatives of the first order:

Let's find the partial derivatives of the second order:

.

Integrals

8.1. Indefinite integral. Detailed solution examples

Let's start studying the topic “ Indefinite integral ", and also consider in detail examples of solutions of the simplest (and not quite) integrals. As usual, we will restrict ourselves to a minimum of theory, which is in numerous textbooks, our task is to learn how to solve integrals.

What do you need to know to successfully master the material? In order to cope with integral calculus, you need to be able to find derivatives, at least at the average level. It will not be a superfluous experience if you have several dozen, or better - a hundred, independently found derivatives behind you. At the very least, you should not be confused by the tasks of differentiating the simplest and most common functions.

It would seem, where are the derivatives here in general, if the article deals with integrals ?! Here's the thing. The fact is that finding derivatives and finding indefinite integrals (differentiation and integration) are two mutually inverse actions, such as addition / subtraction or multiplication / division. Thus, without skill and some kind of experience in finding derivatives, unfortunately, no further progress can be made.

In this regard, we need the following teaching materials: Derivatives tableand Integral table.

What is the difficulty in studying indefinite integrals? If derivatives have strictly 5 rules of differentiation, a table of derivatives and a fairly clear algorithm of actions, then in integrals everything is different. There are dozens of ways and techniques of integration. And, if the integration method was initially chosen incorrectly (ie you do not know how to solve it), then the integral can be “pricked” literally for days, like a real rebus, trying to notice various techniques and tricks. Some even like it.

By the way, we quite often heard from students (not humanities) the opinion like: "I never had an interest in solving the limit or derivative, but integrals are a completely different matter, it's exciting, there is always a desire to" crack "a complex integral" ... Stop. Enough of black humor, let's move on to these indefinite integrals.

Since there are many ways to solve it, where does a teapot start studying indefinite integrals? In integral calculus, in our opinion, there are three pillars or a kind of "axis" around which everything else revolves. First of all, you need to understand well the simplest integrals (this article).

Then you need to work out the lesson in detail. THIS IS THE MOST IMPORTANT RECEPTION! Maybe even the most important article of all articles on integrals. And thirdly, you should definitely familiarize yourself with by integration by partsbecause it integrates a vast class of functions. If you master at least these three lessons, then already "not two". You may be "forgiven" for not knowing integrals of trigonometric functions, integrals of fractions, integrals of rational fractional functions, integrals of irrational functions (roots), but if you "sit in a puddle" on the replacement method or the method of integration by parts - it will be very, very bad.

So let's start simple. Let's look at the table of integrals. As with derivatives, we notice several rules of integration and a table of integrals of some elementary functions. Any tabular integral (and indeed any indefinite integral) has the form:

We immediately understand the notation and terms:

- integral icon.

- integrand function (written with the letter "s").

- differential icon. We will look at what it is very soon. The main thing is that when writing the integral and during the solution, it is important not to lose this icon. There will be a noticeable flaw.

- integrand or "filling" of the integral.

– antiderivativefunction.

– ... You do not need to be heavily loaded with terms, the most important thing here is that in any indefinite integral, a constant is added to the answer.

Solving an indefinite integral means finding set of antiderivativesfrom the given integrand

Let's look at the entry again:

Let's look at the table of integrals.

What's happening? Left parts with us turnto other functions:.

Let's simplify our definition:

Solve indefinite integral - this means TURNING it into an undefined (up to a constant) function , using some rules, techniques and a table.

Take, for example, the tabular integral ... What happened? Symbolic notation has become a multitude of antiderivative functions.

As in the case of derivatives, in order to learn how to find integrals, you do not need to be aware of what an integral is, or an antiderivative function from a theoretical point of view. It is enough to simply carry out transformations according to some formal rules. So, in the case it is not at all necessary to understand why the integral turns into exactly. You can take this and other formulas for granted. Everyone uses electricity, but few people think about how electrons run along the wires there.

Since differentiation and integration are opposite operations, the following is true for any antiderivative that is found directionally:

In other words, if the correct answer is differentiated, then the original integrand must necessarily be obtained.

Let's go back to the same tabular integral .

Let us verify the validity of this formula. We take the derivative from the right side:

Is the original integrand function.

By the way, it became clearer why a constant is always assigned to a function. When differentiating, the constant always turns into zero.

Solve indefinite integral- it means to find a bunch of of allantiderivatives, not any one function. In the considered tabular example,,,, etc. - all these functions are the solution of the integral. There are infinitely many solutions, so they write briefly:

Thus, any indefinite integral is easy enough to check. This is some compensation for a large number of integrals of different types.

Let's move on to considering specific examples. Let's start, as in the study of the derivative, with two rules of integration:

- constant C can (and should) be taken out of the integral sign.

- the integral of the sum (difference) of two functions is equal to the sum (difference) of two integrals. This rule is valid for any number of terms.

As you can see, the rules are basically the same as for derivatives. Sometimes they are called linearity propertiesintegral.

Example 1

Find the indefinite integral.

Check.

Decision: It is more convenient to convert it like.

(1) Apply the rule ... We forget to write down the differential icon dx under each integral. Why under each? dxIs a full multiplier. If you paint in detail, then the first step should be written as follows:

.

(2) According to the rule we move all constants outside the integral signs. Please note that in the last term tg5 is a constant, we also move it out.

In addition, at this step, we prepare the roots and degrees for integration. In the same way as in differentiation, the roots must be represented in the form ... Roots and powers that are in the denominator - move up.

Note: Unlike derivatives, roots in integrals do not always have to be reduced to the form , and move the degrees up.

For instance, - this is a ready-made tabular integral, which has already been counted before you, and all sorts of Chinese tricks like absolutely unnecessary. Similarly: - this is also a tabular integral, it makes no sense to represent the fraction in the form ... Study the table carefully!

(3) All integrals are tabular. We carry out the transformation using a table using the formulas: and

for a power function - .

It should be noted that the tabular integral is a special case of the formula for the power function: .

ConstantC it is enough to add once at the end of the expression

(rather than putting them after each integral).

(4) We write the result obtained in a more compact form when all powers of the form

again represent in the form of roots, and the powers with a negative exponent are reset back to the denominator.

Checking. In order to perform the check, you need to differentiate the received answer:

Received the original integrand, i.e., the integral is found correctly. From what they danced, they returned to that. It's good when the story with the integral ends up like this.

From time to time, there is a slightly different approach to checking the indefinite integral, when the differential is taken from the answer, not the derivative:

.

As a result, we get not an integrand, but an integrand.

Don't be intimidated by the concept of a differential.

The differential is the derivative multiplied by dx.

However, it is not theoretical subtleties that are important to us, but what to do with this differential further. The differential is expanded as follows: icon d we remove, put a stroke on the right above the parenthesis, at the end of the expression we assign a factor dx :

Received the original integrand, that is, the integral is found correctly.

As you can see, the differential is reduced to finding the derivative. I like the second method of checking less, since I have to additionally draw large brackets and drag the differential icon dx until the end of the check. Although it is more correct, or "more solid", or something.

In fact, it was possible to keep silent about the second method of verification. The point is not in the way, but in the fact that we have learned to open the differential. Again.

The differential is revealed as follows:

1) icon d we remove;

2) on the right above the parenthesis we put a stroke (designation of the derivative);

3) at the end of the expression we assign a factor dx .

For instance:

Remember this. We will need this technique very soon.

Example 2

.

When we find an indefinite integral, we ALWAYS try to checkmoreover, there is a great opportunity for this. From this point of view, not all types of problems in higher mathematics are a gift. It doesn't matter that often in control tasks, verification is not required, no one, and nothing prevents it from being carried out on a draft. An exception can be made only when there is not enough time (for example, on a test, an exam). Personally, I always check integrals, and I consider the absence of a check to be a hack and a poorly performed task.

Example 3

Find the indefinite integral:

... Check.

Solution: Analyzing the integral, we see that under the integral we have the product of two functions, and even the exponentiation of an integer expression. Unfortunately, in the field of integral battle no good and comfortable formulas for integrating the product and the quotient as: or .

Therefore, when a product or a quotient is given, it always makes sense to look, but is it possible to transform the integrand into a sum? The example under consideration is the case when it is possible.

First, we will give a complete solution, comments will be below.

(1) We use the good old formula for the square of the sum for any real numbers, getting rid of the degree above the common parenthesis. brackets and applying the formula for abbreviated multiplication in the opposite direction:.

Example 4

Find the indefinite integral

Check.

This is an example for your own solution. Answer and complete solution at the end of the tutorial.

Example 5

Find the indefinite integral

... Check.

In this example, the integrand is a fraction. When we see a fraction in the integrand, the first thought should be the question: "Is it possible to somehow get rid of this fraction, or at least simplify it?"

Notice that the denominator contains the lone root of "x". One in the field is not a warrior, which means that you can divide the numerator by the denominator by term:

We do not comment on actions with fractional powers, since they were repeatedly discussed in articles about the derivative of a function.

If you are still puzzled by an example such as

and no one gets the right answer,

Also note that the solution is missing one step, namely, applying the rules , ... Usually, with some experience in solving integrals, these rules are considered an obvious fact and are not described in detail.

Example 6

Find the indefinite integral. Check.

This is an example for your own solution. Answer and complete solution at the end of the tutorial.

In the general case, it's not so simple with fractions in integrals, additional material on the integration of some types of fractions can be found in the article: Integration of some fractions... But, before moving on to the above article, you need to read the lesson: Replacement method in the indefinite integral... The fact is that the summation of a function under a differential or a method of changing a variable is key pointin the study of the topic, since it occurs not only in "pure problems for the replacement method", but also in many other varieties of integrals.

Solutions and Answers:

Example 2: Solution:

Example 4: Solution:

In this example, we used the abbreviated multiplication formula

Example 6: Solution:

Variable change method in indefinite integral. Solution examples

In this lesson, we will get acquainted with one of the most important and most common techniques used in solving indefinite integrals - the variable change method. To successfully master the material, initial knowledge and integration skills are required. If there is a feeling of an empty full teapot in integral calculus, then you should first familiarize yourself with the material Indefinite integral. Solution examples, where it is explained in an accessible form what an integral is and basic examples for beginners are discussed in detail.

Technically, the method of changing a variable in an indefinite integral is implemented in two ways:

- Bringing the function under the differential sign.

- The actual replacement of the variable.

In fact, they are one and the same, but the design of the solution looks different. Let's start with a simpler case.