Application

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On the base school curriculum this section of mathematics is also studied, but not in detail, but only the basics of such a complex and important topic. In most cases, students begin studying integrals with extensive theory, which is also preceded by important topics, such as the derivative and passage to the limit - they are also limits. Solving integrals gradually begins with the most elementary examples of simple functions, and ends with the use of many approaches and rules proposed in the last century and even much earlier. Integral calculus is for educational purposes in lyceums and schools, that is, in secondary educational institutions. Our website will always help you and solving integrals online will become commonplace for you, and most importantly, an understandable task. Based on this resource, you can easily achieve perfection in this mathematical section. 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To do this, first calculate the indefinite integral according to all the laws of mathematics online. then, if necessary, substitute the upper and lower values ​​of the integral. If you do not need to determine or calculate numeric value indefinite integral, then a constant is added to the resulting antiderivative function, thereby defining a family of antiderivative functions. Special place in science and in general in any engineering field, including continuum mechanics, integration describes entire mechanical systems, their movements and much more. In many cases, the composed integral determines the law of motion material point. It is a very important tool in the study of applied sciences. Based on this, one cannot help but mention large-scale calculations to determine the laws of existence and behavior mechanical systems. Calculator for solving integrals online on the website is a powerful tool for professional engineers. We definitely guarantee this to you, but we will be able to calculate your integral only after you enter the correct expression into the domain of the integrand. Don’t be afraid to make mistakes, everything can be corrected in this matter! Usually, solving integrals comes down to using tabular functions from well-known textbooks or encyclopedias. Like any other indefinite integral, it will be calculated using the standard formula without any major criticisms. First-year students easily and naturally grasp the material they have studied on the spot, and for them finding an integral sometimes takes no more than two minutes. And if a student has learned the table of integrals, then he can generally determine the answers in his head. Expanding functions by variables relative to surfaces initially means the correct vector direction at some abscissa point. The unpredictable behavior of surface lines takes definite integrals as a basis in the response source of mathematical functions. The left edge of the ball does not touch the cylinder in which the circle is inscribed, if you look at the cut in a plane. The sum of small areas divided into hundreds of piecewise continuous functions is the online integral of given function. The mechanical meaning of the integral lies in many applied problems, such as determining the volume of bodies and calculating the mass of a body. Triple and double integrals are involved in these calculations. We insist that the solution of integrals online is carried out only under the supervision of experienced teachers and through numerous checks. We are often asked about the performance of students who do not attend lectures, skip them for no reason, and how they manage to find the integral themselves. We answer that students are free people and are quite capable of studying externally, preparing for a test or exam in the comfort of their own home. In a matter of seconds, our service will help anyone calculate the integral of any given function over a variable. The result obtained should be checked by taking the derivative of the antiderivative function. In this case, the constant from the solution of the integral becomes zero. This rule obviously applies to everyone. As multidirectional operations are justified, the indefinite integral is often reduced to dividing the domain into small parts. However, some students and schoolchildren neglect this requirement. As always, online integrals can be solved in detail by our service website and there are no restrictions on the number of requests, everything is free and available to everyone. There are not many sites that provide a step-by-step answer in a matter of seconds, and most importantly with high accuracy and in a convenient form. In the last example on page five homework I came across one that indicates the need to calculate the integral step by step. But we must not forget about how it is possible to find the integral using a ready-made service, time-tested and tested on thousands of solved examples online. How such an integral determines the motion of the system is clearly and clearly demonstrated to us by the nature of the motion of the viscous fluid, which is described by this system of equations.

Solving integrals is an easy task, but only for a select few. This article is for those who want to learn to understand integrals, but know nothing or almost nothing about them. Integral... Why is it needed? How to calculate it? What are definite and indefinite integrals? If the only use you know of for an integral is to use a crochet hook shaped like an integral icon to get something useful out of hard-to-reach places, then welcome! Find out how to solve integrals and why you can't do without it.

We study the concept of "integral"

Integration was known back in Ancient Egypt. Of course not in modern form, but still. Since then, mathematicians have written many books on this topic. Especially distinguished themselves Newton And Leibniz , but the essence of things has not changed. How to understand integrals from scratch? No way! To understand this topic you will still need basic knowledge basics mathematical analysis. It is this fundamental information that you will find on our blog.

Indefinite integral

Let us have some function f(x) .

Indefinite integral function f(x) this function is called F(x) , whose derivative is equal to the function f(x) .

In other words, the integral is a derivative in reverse or antiderivative. By the way, read about how in our article.

An antiderivative exists for all continuous functions. Also, a constant sign is often added to the antiderivative, since the derivatives of functions that differ by a constant coincide. The process of finding the integral is called integration.

Simple example:

In order not to constantly calculate antiderivatives of elementary functions, it is convenient to summarize them in a table and use ready-made values:

Definite integral

When dealing with the concept of an integral, we are dealing with infinitesimal quantities. The integral will help to calculate the area of ​​the figure, the mass of the inhomogeneous body, the distance traveled at uneven movement path and much more. It should be remembered that an integral is the sum of an infinitely large number of infinitesimal terms.

As an example, imagine a graph of some function. How to find the area of ​​a figure, limited by schedule functions?

Using an integral! Let us divide the curvilinear trapezoid, limited by the coordinate axes and the graph of the function, into infinitesimal segments. This way the figure will be divided into thin columns. The sum of the areas of the columns will be the area of ​​the trapezoid. But remember that such a calculation will give an approximate result. However, the smaller and narrower the segments, the more accurate the calculation will be. If we reduce them to such an extent that the length tends to zero, then the sum of the areas of the segments will tend to the area of ​​the figure. This is a definite integral, which is written like this:

Points a and b are called limits of integration.

Bari Alibasov and the group "Integral"

By the way! For our readers there is now a 10% discount on

Rules for calculating integrals for dummies

Properties of the indefinite integral

How to solve an indefinite integral? Here we will look at the properties of the indefinite integral, which will be useful in solving examples.

• The derivative of the integral is equal to the integrand:

• The constant can be taken out from under the integral sign:

• Integral of the sum equal to the sum integrals. This is also true for the difference:

Properties of a definite integral

• Linearity:

• The sign of the integral changes if the limits of integration are swapped:

• At any points a, b And With:

We have already found out that a definite integral is the limit of a sum. But how to get a specific value when solving an example? For this there is the Newton-Leibniz formula:

Examples of solving integrals

Below we will consider several examples of finding indefinite integrals. We invite you to figure out the intricacies of the solution yourself, and if something is unclear, ask questions in the comments.

To reinforce the material, watch a video about how integrals are solved in practice. Don't despair if the integral is not given right away. Ask and they will tell you everything they know about calculating integrals. With our help, any triple or curved integral over a closed surface will be within your power.

Find indefinite integral(set of antiderivatives or "antiderivatives") means to restore a function from a known derivative of this function. Restored set of antiderivatives F(x) + WITH for function f(x) takes into account the integration constant C. Based on the speed of movement of a material point (derivative), the law of motion of this point (antiderivative) can be restored; according to the acceleration of a point's movement - its speed and the law of motion. As you can see, integration is a wide field for the activities of the Sherlock Holmeses of physics. And in economics, many concepts are represented through functions and their derivatives, and therefore, for example, it is possible to restore the volume of products produced at the corresponding time using labor productivity at a certain point in time (derivative).

To find the indefinite integral, it takes quite a bit a large number of basic integration formulas. But the process of finding it is much more difficult than just applying these formulas. All the complexity does not relate to integration, but to bringing the integrable expression to a form that makes it possible to find the indefinite integral using the basic formulas mentioned above. This means that to begin the practice of integration, you need to activate what you have learned in high school expression transformation skills.

We will learn to find integrals using properties and table of indefinite integrals from a lesson about the basic concepts of this topic (opens in a new window).

There are several methods for finding the integral, of which variable replacement method And integration by parts method- a mandatory gentleman's set for everyone who has successfully passed higher mathematics. However, it is more useful and enjoyable to start mastering integration using the expansion method, based on the following two theorems on the properties of the indefinite integral, which we repeat here for convenience.

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

Theorem 4. The indefinite integral of an algebraic sum of a finite number of functions is equal to algebraic sum indefinite integrals of these functions, i.e.

(2)

In addition, the following rule may be useful in integration: if the expression of the integrand contains a constant factor, then the expression of the antiderivative is multiplied by the inverse of the constant factor, that is

(3)

Since this is an introductory lesson to solving integration problems, it is important to note two things that either at the very beginning or a little later may surprise you. The surprise is due to the fact that integration is the inverse operation of differentiation and the indefinite integral can rightly be called the “antiderivative”.

The first thing you shouldn't be surprised by when integrating. In the table of integrals there are formulas that have no analogues among the derivative table formulas . These are the following formulas:

However, you can make sure that the derivatives of the expressions on the right sides of these formulas coincide with the corresponding integrands.

The second thing that should not be surprising when integrating. Although the derivative of any elementary function is also an elementary function, indefinite integrals of some elementary functions are no longer elementary functions . Examples of such integrals could be the following:

To develop integration techniques, the following skills will be useful: reducing fractions, dividing a polynomial in the numerator of a fraction by a monomial in the denominator (to obtain the sum of indefinite integrals), converting roots to powers, multiplying a monomial by a polynomial, raising to a power. These skills are needed for transformations of the integrand, which should result in the sum of the integrals present in the table of integrals.

Finding indefinite integrals together

Example 1. Find the indefinite integral

.

Solution. We see in the denominator of the integrand a polynomial in which x is squared. This is an almost sure sign that you can apply table integral 21 (with an arctangent as a result). We take out the factor-two from the denominator (there is such a property of the integral - the constant factor can be taken out beyond the sign of the integral; it was mentioned above as Theorem 3). The result of all this:

Now the denominator is the sum of squares, which means that we can apply the mentioned table integral. Finally we get the answer:

.

Example 2. Find the indefinite integral

Solution. We again apply Theorem 3 - the property of the integral, on the basis of which the constant factor can be taken out of the sign of the integral:

We apply formula 7 from the table of integrals (variable to power) to the integrand function:

.

We reduce the resulting fractions and we have the final answer:

Example 3. Find the indefinite integral

Solution. Applying first Theorem 4 and then Theorem 3 on properties, we find this integral as the sum of three integrals:

All three obtained integrals are tabular. We use formula (7) from the table of integrals for n = 1/2, n= 2 and n= 1/5, and then

combines all three arbitrary constants that were introduced when finding the three integrals. Therefore, in similar situations, only one arbitrary integration constant should be introduced.

Example 4. Find the indefinite integral

Solution. When the denominator of the integrand contains a monomial, we can divide the numerator by the denominator term by term. The original integral turned into the sum of two integrals:

.

To apply the table integral, we transform the roots into powers and here is the final answer:

We continue to find indefinite integrals together

Example 7. Find the indefinite integral

Solution. If we transform the integrand by squaring the binomial and dividing the numerator by the denominator term by term, then the original integral becomes the sum of three integrals.

Indefinite integral.
Detailed examples solutions

In this lesson we will begin to study the topic Indefinite integral, and we will also analyze in detail examples of solutions to the simplest (and not so simple) integrals. In this article I will limit myself to a minimum of theory, and now 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 a minimum, at an intermediate level. Therefore, if the material has been launched, I recommend that you first carefully read the lessons How to find the derivative? And Derivative of a complex function. It will not be a waste of experience if you have several dozen (preferably a hundred) independently found derivatives under your belt. At the very least, you should not be confused by tasks to differentiate the simplest and most common functions. It would seem, what do derivatives have to do with it if the article is about 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 the skill (+ some experience) of finding derivatives, unfortunately, you cannot move forward.

In this regard, we will need the following teaching materials: Derivatives table And Table of integrals. Reference manuals can be opened, downloaded or printed on the page Mathematical formulas and tables.

What is the difficulty in learning indefinite integrals? If in derivatives there are 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 integration methods and techniques. And, if the integration method is initially chosen incorrectly (i.e. you don’t know how to solve), then the integral can be “pricked” literally for days, like a real puzzle, trying to spot various techniques and tricks. Some people even like it. By the way, this is not a joke, I quite often heard from students an opinion like “I have never had any interest in solving a limit or derivative, but integrals are a completely different matter, it’s fascinating, there is always a desire to “hack” a complex integral.” Stop. Enough of the black humor, let's move on to these very indefinite integrals.

Since there are so many ways to solve it, then where to start studying indefinite integrals for a teapot? In integral calculus, in my opinion, there are three pillars or a kind of “axis” around which everything else revolves. First of all, you should have a good understanding of the simplest integrals (this article). Then you need to work through the lesson in detail. THIS THE MOST IMPORTANT TECHNIQUE! Perhaps even the most important article of all my articles on integrals. And thirdly, you should definitely familiarize yourself with the method of integration by parts, since it can be used to integrate a wide class of functions. If you master at least these three lessons, then you will no longer have two. You may be forgiven for not knowing integrals from trigonometric functions, integrals from fractions, integrals from fractional-rational functions, integrals from irrational functions (roots), but if you get stuck on the replacement method or the method of integration by parts, then it will be very, very bad.

Demotivators are now very common on the RuNet. In the context of studying integrals, on the contrary, it is simply necessary MOTIVATOR. Like in that joke about Vasily Ivanovich, who motivated both Petka and Anka. Dear lazy people, freeloaders and other normal students, be sure to read the following. Knowledge and skills on the indefinite integral will be required in further studies, in particular, when studying the definite integral, improper integrals, and differential equations in the 2nd year. The need to take the integral arises even in probability theory! Thus, without integrals, the path to the summer session and 2nd year WILL BE REALLY CLOSED. I'm serious. The conclusion is this. The more integrals various types you decide, the easier it will be future life . Yes, it will take quite a lot of time, yes, sometimes you don’t want to, yes, sometimes “to hell with it, with this integral, maybe I won’t get it.” But the next thought should inspire and warm your soul; your efforts will pay off in full! You will be able to crack differential equations like nuts and easily deal with integrals that you will encounter in other sections of higher mathematics. Having thoroughly understood the indefinite integral, YOU WILL ACTUALLY MASTER A SEVERAL MORE SECTIONS OF THE TOWER.

And so I just couldn't help but create intensive course on the technique of integration, which turned out to be surprisingly short - those who wish can use the pdf book and prepare VERY quickly. But the materials on the site are by no means worse!

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

Let’s immediately understand the notations and terms:

– integral icon.

– integrand function (written with the letter “s”).

– differential icon. When writing the integral and during the solution, it is important not to lose this icon. There will be a noticeable flaw.

– integrand expression or “filling” of the integral.

– antiderivative function.

– many original functions. There is no need to be heavily loaded with terms; the most important thing is that in any indefinite integral a constant is added to the answer.

Solving an integral means finding a certain function using some rules, techniques and a table.

Let's look at the entry again:

Let's look at the table of integrals.

What's happening? We have the left parts turn into to other functions: .

Let's simplify our definition.

Solving an indefinite integral means TRANSFORMING it into a definite function, using some rules, techniques and a table.

Take, for example, the table integral . What happened? turned into a function.

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

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

In other words, if you differentiate the correct answer, then you must get the original integrand function.

Let's return to the same table integral .

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

is the original integrand function.

By the way, it has become clearer why a constant is always assigned to a function. When differentiated, the constant always turns to zero.

Solve indefinite integral- it means to find a bunch of everyone antiderivatives, and not just one function. In the table example under consideration, , , , etc. – all these functions are solutions to the integral. There are infinitely many solutions, so we write it down briefly:

Thus, any indefinite integral is quite easy to check (unlike derivatives, where a good check can only be done using mathematical programs). This is some compensation for a large number of integrals of different types.

Let's move on to consider specific examples. Let's start, as in studying the derivative,
with two rules of integration, also called linearity properties indefinite integral:

– the constant factor can (and should) be taken out of the integral sign.

– the integral of the algebraic sum of two functions is equal to the algebraic sum of two integrals of each function separately. This property valid for any number of terms.

As you can see, the rules are basically the same as for derivatives.

Example 1

Solution: It is more convenient to rewrite it on paper.

(1) Apply the rule . Don't forget to write down the differential symbol under each integral. Why under each? - this is a full multiplier, if we describe the solution in detail, then the first step should be written like this:

(2) According to the rule , we take all the constants outside the integral signs. Please note that the last term is a constant, we also take it out.
In addition, at this step we prepare roots and powers for integration. In the same way as with differentiation, the roots must be represented in the form . Move the roots and powers that are located in the denominator upward.

! Note: unlike derivatives, roots in integrals should not always be reduced to the form , but degrees should be transferred upward. For example, this is a ready-made table integral, and all sorts of Chinese tricks like completely unnecessary. Similarly: – also a tabular integral, there is no point in representing the fraction in the form . Study the table carefully!

(3) All our integrals are tabular. We carry out the transformation using a table using the formulas: , And .
Special attention I refer to the integration formula power function , it occurs very often, it is better to remember it. It should be noted that the table integral is special case the same formula: .
It is enough to add the constant once at the end of the expression (and not put them after each integral).
(4) We write the result obtained in a more compact form, all powers of the form are again represented as roots, powers with a negative exponent are reset back to the denominator.

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

Received the original integrand, which means the integral was found correctly. What they danced from is what they returned to. You know, it’s very good when a story with an integral ends this way.

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

Those who understood from the first semester understood, but now what is important to us is not the theoretical subtleties, but what is important is what to do next with this differential. It needs to be revealed, and from a formal technical point of view, this is almost the same as finding a derivative. The differential is revealed as follows: we remove the icon, put a stroke on the right above the bracket, and add a factor to the end of the expression:

Received original integrand, which means the integral was found correctly.

I like the second method of checking less, since I have to additionally draw large brackets and drag the differential icon until the end of the check. Although it is more correct or “more respectable” or something.

In fact, I could have kept silent about the second verification method altogether. The point is not in the method, but in the fact that we have learned to open the differential. Again.

The differential is revealed as follows:

1) remove the icon;
2) on the right above the bracket we put a stroke (denotation of the derivative);
3) at the end of the expression we assign a factor .

For example:

Remember this. We will need this technique very soon.

Example 2

Find the indefinite integral. Perform check.

When we find an indefinite integral, we ALWAYS try to check Moreover, there is a great opportunity for this. Not all types of problems in higher mathematics are a gift from this point of view. It doesn't matter that often test assignments no verification is required, no one is checking it, 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, during a test or exam). Personally, I always check integrals, and I consider the lack of checking to be a hack job and a poorly completed task.

Example 3

Find the indefinite integral. Perform check.

Solution: Analyzing the integral, we see that we have the product of two functions, and even the exponentiation of an entire expression. Unfortunately, in the field of integral battle there are no good and convenient formulas for integrating the product and the particular , .

And therefore, when a product or quotient is given, it always makes sense to see if it is possible to transform the integrand into a sum?

The example under consideration is the case when it is possible. First I will give the complete solution, comments will be below.

(1) We use the good old formula of the square of the sum, getting rid of the degree.

(2) We put it in brackets, getting rid of the product.

Example 4

Find the indefinite integral. Perform check.

This is an example for you to solve yourself. The answer and complete solution are at the end of the lesson.

Example 5

Find the indefinite integral. Perform check.

IN 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?

We notice that the denominator contains a single root of “X”. One in the field is not a warrior, which means we can divide the numerator by the denominator term by term:

Actions with fractional powers I do not comment, since they have been discussed repeatedly in articles on the derivative function. If you are still perplexed by an example like , and you still can’t get the correct answer, then I recommend turning to school textbooks. In higher mathematics, fractions and operations with them are encountered at every step.

Also note that the solution is missing one step, namely applying the rules , . Usually, even during the initial experience of solving integrals, these properties are taken for granted and are not described in detail.

Example 6

Find the indefinite integral. Perform check.

This is an example for you to solve yourself. The answer and complete solution are at the end of the lesson.

In general, things are not so simple with fractions in integrals, additional material on the integration of fractions of some types can be found in the article Integrating Some Fractions.

! But, before moving on to the above article, you need to familiarize yourself with the lesson Substitution method in indefinite integral. The point is that subsuming a function under a differential or variable replacement method is key point in the study of the topic, since it is found not only “in pure tasks on the replacement method,” but also in many other types of integrals.

I really wanted to include a few more examples in this lesson, but I’m sitting here now, typing this text in Verde and noticing that the article has already grown to a decent size.
And therefore introductory course integrals for dummies has come to an end.

I wish you success!

Solutions and answers:

Example 2: Solution:

Example 4: Solution:

In this example we used the abbreviated multiplication formula

Example 6: Solution:

I completed the check, and you? ;)