Definite integral solution algorithm. Solving a definite integral online

Definite integral. Solution examples

Hello again. In this lesson, we will analyze in detail such a wonderful thing as a definite integral. This time the introduction will be short. Everything. Because a snowstorm outside the window.

In order to learn how to solve certain integrals, you need to:

1) be able find indefinite integrals.

2) be able calculate definite integral.

As you can see, in order to master the definite integral, you need to be fairly well versed in the "ordinary" indefinite integrals. Therefore, if you are just starting to dive into integral calculus, and the kettle has not yet boiled at all, then it is better to start with the lesson Indefinite integral. Solution examples. In addition, there are pdf courses for ultrafast training- if you have literally a day, half a day left.

In general, the definite integral is written as:

What has been added compared to the indefinite integral? added integration limits.

Lower limit of integration
Upper limit of integration standardly denoted by the letter .
The segment is called segment of integration.

Before we move on to practical examples, a small faq on the definite integral.

What does it mean to solve a definite integral? Solving a definite integral means finding a number.

How to solve a definite integral? With the help of the Newton-Leibniz formula familiar from school:

It is better to rewrite the formula on a separate piece of paper; it should be in front of your eyes throughout the lesson.

The steps for solving a definite integral are as follows:

1) First we find the antiderivative function (indefinite integral). Note that the constant in the definite integral not added. The designation is purely technical, and the vertical stick does not carry any mathematical meaning, in fact it is just a strikethrough. Why is the record necessary? Preparation for applying the Newton-Leibniz formula.

2) We substitute the value of the upper limit in the antiderivative function: .

3) We substitute the value of the lower limit into the antiderivative function: .

4) We calculate (without errors!) the difference, that is, we find the number.

Does a definite integral always exist? No not always.

For example, the integral does not exist because the integration interval is not included in the domain of the integrand (values under the square root cannot be negative). Here's a less obvious example: . Here, on the integration interval tangent endures endless breaks at the points , , and therefore such a definite integral also does not exist. By the way, who has not yet read the methodological material Graphs and basic properties of elementary functions- Now is the time to do it. It will be great to help throughout the course of higher mathematics.

For for the definite integral to exist at all, it is sufficient that the integrand be continuous on the interval of integration.

From the above, the first important recommendation follows: before proceeding with the solution of ANY definite integral, you need to make sure that the integrand continuous on the integration interval. As a student, I repeatedly had an incident when I suffered for a long time with finding a difficult primitive, and when I finally found it, I puzzled over one more question: “what kind of nonsense turned out?”. In a simplified version, the situation looks something like this:

???! You can not substitute negative numbers under the root! What the hell?! initial carelessness.

If for a solution (in a test, in a test, an exam) you are offered an integral like or , then you need to give an answer that this definite integral does not exist and justify why.

! Note : in the latter case, the word "certain" cannot be omitted, because the integral with point discontinuities is divided into several, in this case, into 3 improper integrals, and the formulation "this integral does not exist" becomes incorrect.

Can the definite integral be equal to a negative number? Maybe. And a negative number. And zero. It may even turn out to be infinity, but it will already be improper integral, which is given a separate lecture.

Can the lower limit of integration be greater than the upper limit of integration? Perhaps such a situation actually occurs in practice.

- the integral is calmly calculated using the Newton-Leibniz formula.

What does higher mathematics not do without? Of course, without all sorts of properties. Therefore, we consider some properties of a definite integral.

In a definite integral, you can rearrange the upper and lower limits, while changing the sign:

For example, in a definite integral before integration, it is advisable to change the limits of integration to the "usual" order:

- in this form, integration is much more convenient.

- this is true not only for two, but also for any number of functions.

In a definite integral, one can carry out change of integration variable, however, in comparison with the indefinite integral, this has its own specifics, which we will talk about later.

For a definite integral, formula for integration by parts:

Example 1

Decision:

(1) We take the constant out of the integral sign.

(2) We integrate over the table using the most popular formula . It is advisable to separate the appeared constant from and put it out of the bracket. It is not necessary to do this, but it is desirable - why extra calculations?

. First we substitute in the upper limit, then the lower limit. We carry out further calculations and get the final answer.

Example 2

Calculate a definite integral

This is an example for self-solving, solution and answer at the end of the lesson.

Let's make it a little more difficult:

Example 3

Calculate a definite integral

Decision:

(1) We use the linearity properties of the definite integral.

(2) We integrate over the table, while taking out all the constants - they will not participate in the substitution of the upper and lower limits.

(3) For each of the three terms, we apply the Newton-Leibniz formula:

A WEAK LINK in a definite integral is calculation errors and a common SIGN CONFUSION. Be careful! I focus on the third term: - first place in the hit parade of mistakes due to inattention, very often they write automatically (especially when the substitution of the upper and lower limits is carried out orally and is not signed in such detail). Once again, carefully study the above example.

It should be noted that the considered method of solving a definite integral is not the only one. With some experience, the solution can be significantly reduced. For example, I myself used to solve such integrals like this:

Here I verbally used the rules of linearity, orally integrated over the table. I ended up with just one parenthesis with the limits outlined: (as opposed to the three brackets in the first method). And in the "whole" antiderivative function, I first substituted 4 first, then -2, again doing all the actions in my mind.

What are the disadvantages of the short solution method? Everything is not very good here from the point of view of the rationality of calculations, but personally I don’t care - I count ordinary fractions on a calculator.
In addition, there is an increased risk of making a mistake in the calculations, so it is better for a student-dummies to use the first method, with “my” solution method, the sign will definitely be lost somewhere.

However, the undoubted advantages of the second method are the speed of the solution, the compactness of the notation, and the fact that the antiderivative is in one bracket.

Tip: before using the Newton-Leibniz formula, it is useful to check: has the antiderivative itself been found correctly?

So, in relation to the example under consideration: before substituting the upper and lower limits into the antiderivative function, it is advisable to check on a draft whether the indefinite integral was found correctly at all? Differentiate:

The original integrand was obtained, which means that the indefinite integral was found correctly. Now you can apply the Newton-Leibniz formula.

Such a check will not be superfluous when calculating any definite integral.

Example 4

Calculate a definite integral

This is an example for self-solving. Try to solve it in a short and detailed way.

Change of variable in a definite integral

For the definite integral, all types of substitutions are valid, as for the indefinite integral. Thus, if you are not very good at substitutions, you should carefully read the lesson. Replacement method in indefinite integral.

There is nothing scary or complicated about this paragraph. The novelty lies in the question how to change the limits of integration when replacing.

In the examples, I will try to give such types of replacements that have not yet been seen anywhere on the site.

Example 5

Calculate a definite integral

The main question here is not at all in a definite integral, but how to correctly carry out the replacement. We look in integral table and we figure out what our integrand most of all looks like? Obviously, on the long logarithm: . But there is one inconsistency, in the tabular integral under the root, and in ours - "x" to the fourth degree. The idea of replacement follows from the reasoning - it would be nice to somehow turn our fourth power into a square. This is real.

First, we prepare our integral for replacement:

From the above considerations, the replacement naturally suggests itself:
Thus, everything will be fine in the denominator: .
We find out what the rest of the integrand will turn into, for this we find the differential:

Compared to the replacement in the indefinite integral, we add an additional step.

Finding new limits of integration.

It's simple enough. We look at our replacement and the old limits of integration , .

First, we substitute the lower limit of integration, that is, zero, into the replacement expression:

Then we substitute the upper limit of integration into the replacement expression, that is, the root of three:

Ready. And just something…

Let's continue with the solution.

(1) According to replacement write a new integral with new limits of integration.

(2) This is the simplest table integral, we integrate over the table. It is better to leave the constant outside the brackets (you can not do this) so that it does not interfere in further calculations. On the right, we draw a line indicating the new limits of integration - this is preparation for applying the Newton-Leibniz formula.

(3) We use the Newton-Leibniz formula .

We strive to write the answer in the most compact form, here I used the properties of logarithms.

Another difference from the indefinite integral is that, after we have made the substitution, no replacements are required.

And now a couple of examples for an independent solution. What replacements to carry out - try to guess on your own.

Example 6

Calculate a definite integral

Example 7

Calculate a definite integral

These are self-help examples. Solutions and answers at the end of the lesson.

And at the end of the paragraph, a couple of important points, the analysis of which appeared thanks to the site visitors. The first one concerns legitimacy of replacement. In some cases, it cannot be done! So Example 6 would seem to be resolvable with universal trigonometric substitution, but the upper limit of integration ("pi") not included in domain this tangent and therefore this substitution is illegal! Thus, the "replacement" function must be continuous in all points of the segment of integration.

In another e-mail, the following question was received: “Do we need to change the limits of integration when we bring the function under the differential sign?”. At first I wanted to “shrug off the nonsense” and automatically answer “of course not”, but then I thought about the reason for such a question and suddenly discovered that the information lacks. But it is, albeit obvious, but very important:

If we bring the function under the sign of the differential, then there is no need to change the limits of integration! Why? Because in this case no actual transition to new variable. For example:

And here the summing is much more convenient than the academic replacement with the subsequent "painting" of new limits of integration. Thus, if the definite integral is not very complicated, then always try to bring the function under the sign of the differential! It's faster, it's more compact, and it's common - as you will see dozens of times!

Thank you very much for your letters!

Method of integration by parts in a definite integral

There is even less novelty here. All postings of the article Integration by parts in the indefinite integral are fully valid for a definite integral as well.
Plus, there is only one detail, in the formula for integration by parts, the limits of integration are added:

The Newton-Leibniz formula must be applied twice here: for the product and, after we take the integral.

For example, I again chose the type of integral that I have not seen anywhere else on the site. The example is not the easiest, but very, very informative.

Example 8

Calculate a definite integral

We decide.

Integrating by parts:

Who had difficulty with the integral, take a look at the lesson Integrals of trigonometric functions, where it is discussed in detail.

(1) We write the solution in accordance with the formula for integration by parts.

(2) For the product, we use the Newton-Leibniz formula. For the remaining integral, we use the properties of linearity, dividing it into two integrals. Don't get confused by signs!

(4) We apply the Newton-Leibniz formula for the two antiderivatives found.

To be honest, I don't like the formula and, if possible, ... do without it at all! Consider the second way of solving, from my point of view it is more rational.

Calculate a definite integral

In the first step, I find the indefinite integral:

Integrating by parts:

An antiderivative function has been found. It makes no sense to add a constant in this case.

What is the advantage of such a trip? There is no need to “drag along” the limits of integration, indeed, you can be tormented a dozen times by writing small icons of the limits of integration

In the second step, I check(usually on draft).

It's also logical. If I found the antiderivative function incorrectly, then I will also solve the definite integral incorrectly. It is better to find out immediately, let's differentiate the answer:

The original integrand has been obtained, which means that the antiderivative function has been found correctly.

The third stage is the application of the Newton-Leibniz formula:

And there is a significant benefit here! In “my” way of solving, there is a much lower risk of getting confused in substitutions and calculations - the Newton-Leibniz formula is applied only once. If the kettle solves a similar integral using the formula (the first way), then stopudovo will make a mistake somewhere.

The considered solution algorithm can be applied to any definite integral.

Dear student, print and save:

What to do if a definite integral is given that seems complicated or it is not immediately clear how to solve it?

1) First we find the indefinite integral (antiderivative function). If at the first stage there was a bummer, it is pointless to rock the boat with Newton and Leibniz. There is only one way - to increase your level of knowledge and skills in solving indefinite integrals.

2) We check the found antiderivative function by differentiation. If it is found incorrectly, the third step will be a waste of time.

3) We use the Newton-Leibniz formula. We carry out all calculations EXTREMELY CAREFULLY - here is the weakest link in the task.

And, for a snack, an integral for an independent solution.

Example 9

Calculate a definite integral

The solution and the answer are somewhere nearby.

The following recommended tutorial on the topic is − How to calculate the area of a figure using the definite integral?
Integrating by parts:

Did you definitely solve them and get such answers? ;-) And there is porn on the old woman.

In order to learn how to solve certain integrals, you need to:

1) be able find indefinite integrals.

2) be able calculate definite integral.

In general, the definite integral is written as:

What has been added compared to the indefinite integral? added integration limits.

Lower limit of integration
Upper limit of integration standardly denoted by the letter .
The segment is called segment of integration.

Before we move on to practical examples, a little "fuck" on the definite integral.

What is a definite integral? I could tell you about the diameter of the division of the segment, the limit of integral sums, etc., but the lesson is of a practical nature. Therefore, I will say that the definite integral is a NUMBER. Yes, yes, the most common number.

Does the definite integral have a geometric meaning? There is. And very good. The most popular task calculating the area using a definite integral.

What does it mean to solve a definite integral? Solving a definite integral means finding a number.

How to solve a definite integral? With the help of the Newton-Leibniz formula familiar from school:

It is better to rewrite the formula on a separate piece of paper; it should be in front of your eyes throughout the lesson.

The steps for solving a definite integral are as follows:

1) First we find the antiderivative function (indefinite integral). Note that the constant in the definite integral never added. The designation is purely technical, and the vertical stick does not carry any mathematical meaning, in fact it is just a strikethrough. Why is the record necessary? Preparation for applying the Newton-Leibniz formula.

2) We substitute the value of the upper limit in the antiderivative function: .

3) We substitute the value of the lower limit into the antiderivative function: .

4) We calculate (without errors!) the difference, that is, we find the number.

Does a definite integral always exist? No not always.

For example, the integral does not exist because the integration interval is not included in the domain of the integrand (values under the square root cannot be negative). Here's a less obvious example: . Such an integral also does not exist, since there is no tangent at the points of the segment. By the way, who has not yet read the methodological material Graphs and basic properties of elementary functions- Now is the time to do it. It will be great to help throughout the course of higher mathematics.

In order for a definite integral to exist at all, it is necessary that the integrand be continuous on the integration interval.

???!!!

You can not substitute negative numbers under the root!

If for a solution (in a test, in a test, an exam) you are offered a non-existent integral like

then you need to give an answer that the integral does not exist and justify why.

Can the lower limit of integration be greater than the upper limit of integration? Perhaps such a situation actually occurs in practice.

- the integral is calmly calculated using the Newton-Leibniz formula.

What does higher mathematics not do without? Of course, without all sorts of properties. Therefore, we consider some properties of a definite integral.

In a definite integral, you can rearrange the upper and lower limits, while changing the sign:

For example, in a definite integral before integration, it is advisable to change the limits of integration to the "usual" order:

- in this form, integration is much more convenient.

As for the indefinite integral, the linearity properties are valid for the definite integral:

- this is true not only for two, but also for any number of functions.

In a definite integral, one can carry out change of integration variable, however, in comparison with the indefinite integral, this has its own specifics, which we will talk about later.

For a definite integral, formula for integration by parts:

Example 1

Decision:

(1) We take the constant out of the integral sign.

(3) We use the Newton-Leibniz formula

First we substitute in the upper limit, then the lower limit. We carry out further calculations and get the final answer.

Example 2

Calculate a definite integral

This is an example for self-solving, solution and answer at the end of the lesson.

Let's make it a little more difficult:

Example 3

Calculate a definite integral

Decision:

(1) We use the linearity properties of the definite integral.

(2) We integrate over the table, while taking out all the constants - they will not participate in the substitution of the upper and lower limits.

(3) For each of the three terms, we apply the Newton-Leibniz formula:

A WEAK LINK in a definite integral is calculation errors and a common SIGN CONFUSION. Be careful! I focus on the third term:

- first place in the hit parade of mistakes due to inattention, very often they write automatically

(especially when the substitution of the upper and lower limits is carried out orally and is not signed in such detail). Once again, carefully study the above example.

Here I verbally used the rules of linearity, orally integrated over the table. I ended up with just one parenthesis with the limits outlined:

(as opposed to the three brackets in the first method). And in the "whole" antiderivative function, I first substituted 4 first, then -2, again doing all the actions in my mind.

The undoubted advantages of the second method are the speed of the solution, the compactness of the notation, and the fact that the antiderivative

is in one parenthesis.

The process of solving integrals in science called "mathematics" is called integration. With the help of integration, you can find some physical quantities: area, volume, mass of bodies, and much more.

Integrals are indefinite and definite. Consider the form of a definite integral and try to understand its physical meaning. It appears as follows: $$ \int ^a _b f(x) dx $$. A distinctive feature of writing a definite integral from an indefinite one is that there are limits of integration a and b. Now we will find out what they are for, and what a definite integral means. In a geometric sense, such an integral is equal to the area of the figure bounded by the curve f(x), lines a and b, and the Ox axis.

It can be seen from Fig. 1 that the definite integral is the very area that is shaded in gray. Let's check it out with a simple example. Let's find the area of the figure in the image below using integration, and then calculate it in the usual way by multiplying the length by the width.

Figure 2 shows that $ y=f(x)=3 $, $ a=1, b=2 $. Now we substitute them into the definition of the integral, we get that $$ S=\int _a ^b f(x) dx = \int _1 ^2 3 dx = $$ $$ =(3x) \Big|_1 ^2=(3 \ cdot 2)-(3 \cdot 1)=$$ $$=6-3=3 \text(unit)^2 $$ Let's check in the usual way. In our case, length = 3, shape width = 1. $$ S = \text(length) \cdot \text(width) = 3 \cdot 1 = 3 \text(unit)^2 $$ As you can see, everything matched perfectly .

The question arises: how to solve indefinite integrals and what is their meaning? The solution of such integrals is the finding of antiderivative functions. This process is the opposite of finding the derivative. In order to find the antiderivative, you can use our help in solving problems in mathematics, or you must independently memorize the properties of integrals and the integration table of the simplest elementary functions. Finding looks like this $$ \int f(x) dx = F(x) + C \text(where) F(x) $ is the antiderivative of $ f(x), C = const $.

To solve the integral, you need to integrate the function $ f(x) $ with respect to the variable. If the function is tabular, then the answer is written in the appropriate form. If not, then the process is reduced to obtaining a table function from the function $ f(x) $ by tricky mathematical transformations. There are various methods and properties for this, which we will discuss below.

So, now let's make an algorithm how to solve integrals for dummies?

Algorithm for calculating integrals

Find out the definite integral or not.
If undefined, then you need to find the antiderivative function $ F(x) $ of the integrand $ f(x) $ using mathematical transformations that bring the function $ f(x) $ to a tabular form.
If defined, then step 2 must be performed, and then substitute the limits of $a$ and $b$ into the antiderivative function $F(x)$. By what formula to do this, you will learn in the article "Newton Leibniz's Formula".

Solution examples

So, you have learned how to solve integrals for dummies, examples of solving integrals have been sorted out on the shelves. They learned their physical and geometric meaning. Solution methods will be discussed in other articles.

Examples of calculating indefinite integrals

Table Integral Calculation

Substitution integration:

Examples of calculating integrals

Newton–Leibniz basic formula

Substitution calculations

Chapter 4 Differential Equations.

differential equation called an equation that relates an independent variable X , the desired function at and its derivatives or differentials.

The symbolically differentiated equation is written as follows:

The differential equation is called ordinary if the desired function depends on one independent variable.

order differential equation is called the order of the highest derivative (or differential) included in this equation.

Decision(or integral) of a differential equation is a function that turns this equation into an identity.

General solution(or common integral) of a differential equation is a solution that includes as many independent arbitrary constants as the order of the equation. Thus, the general solution of a first-order differential equation contains one arbitrary constant.

Private decision A differential equation is a solution obtained from a general one for various numerical values of arbitrary constants. The values of arbitrary constants are found at certain initial values of the argument and function.

The graph of a particular solution of a differential equation is called integral curve.

The general solution of the differential equation corresponds to the set (family) of all integral curves.

First order differential equation an equation is called, which includes derivatives (or differentials) not higher than the first order.

Differential equation with separable variables is called an equation of the form

To solve this equation, you must first separate the variables:

and then integrate both parts of the resulting equality:

1. Find a general solution to the equation

o Dividing the variables, we have

Integrating both parts of the resulting equation:

Since an arbitrary constant With can take any numerical values, then for the convenience of further transformations instead of C we wrote (1/2) ln C. Potentiating the last equality, we obtain

This is the general solution of this equation.

Literature

V. G. Boltyansky, What is differentiation, "Popular lectures on mathematics",

Issue 17, Gostekhizdat 1955, 64 pp.

V. A. Gusev, A. G. Mordkovich "Mathematics"

G. M. Fikhtengolts "Course of differential and integral calculus", volume 1

V. M. Borodikhin, Higher mathematics, textbook. manual, ISBN 5-7782-0422-1.

Nikolsky SM Chapter 9. Riemann's Definite Integral // Course of Mathematical Analysis. - 1990. - T. 1.

Ilyin V. A., Poznyak, E. G. Chapter 6. Indefinite integral // Fundamentals of Mathematical Analysis. - 1998. - V. 1. - (Course of higher mathematics and mathematical physics).

Demidovich B.P. Department 3. Indefinite integral // Collection of problems and exercises in mathematical analysis. - 1990. - (Course of higher mathematics and mathematical physics).

Valutse I.I., Diligul G.D. Mathematics for technical schools based on secondary school: Textbook-2nd ed.rev. and additional M.6 Science. 1989

Kolyagin Yu.M. Yakovlev G.N. mathematics for technical schools. Algebra and the Beginnings of Analysis Parts 1 and 2. Publishing house "Naukka" M., 1981.

Shchipachev V.S. Tasks in higher mathematics: Proc. Allowance for universities. Higher School 1997

Bogomolov N.V. Practical lessons in mathematics: textbook. Allowance for technical schools. Higher School 1997