How to find a definite integral examples. Definite integral

In order to learn how to solve definite integrals you need to:

1) Be able to find indefinite integrals.

2) Be able to calculate definite integral.

As you can see, in order to master a definite integral, you need to have a fairly good understanding of “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. Examples of solutions.

In general form, the definite integral is written as follows:

What is added compared to the indefinite integral? More limits of integration.

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

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

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

Does the definite integral have geometric meaning? Eat. And very good. The most popular task is calculating 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? Using 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 entire 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 marking. Why is the recording itself needed? Preparation for applying the Newton-Leibniz formula.

2) Substitute the value of the upper limit into the antiderivative function: .

3) 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 segment of integration is not included in the domain of definition 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 hasn’t read the teaching material yet? Graphs and basic properties of elementary functions– the time to do it is now. 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 function be continuous on the interval of integration.

From the above, the first important recommendation follows: before you begin solving ANY definite integral, you need to make sure that the integrand function is continuous on the interval of integration. When I was a student, I repeatedly had an incident when I struggled for a long time with finding a difficult antiderivative, and when I finally found it, I racked my brains over another question: “What kind of nonsense did it turn out to be?” In a simplified version, the situation looks something like this:

???!!!

You cannot substitute negative numbers under the root!

If for a solution (in a test, test, 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 a 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 are given a separate lecture.

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

– the integral can be easily calculated using the Newton-Leibniz formula.

What is higher mathematics indispensable? Of course, without all sorts of properties. Therefore, let us consider some properties of the definite integral.

In a definite integral, you can rearrange the upper and lower limits, 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 it is much more convenient to integrate.

As with the indefinite integral, the definite integral has linear properties:

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

In a definite integral one can carry out replacement of integration variable, however, compared to the indefinite integral, this has its own specifics, which we will talk about later.

For a definite integral the following holds true: integration by parts formula:

Example 1

Solution:

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

(2) Integrate over the table using the most popular formula . It is advisable to separate the emerging constant from and put it outside the bracket. It is not necessary to do this, but it is advisable - why the extra calculations?

(3) We use the Newton-Leibniz formula

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

Example 2

Calculate definite integral

This is an example for you to solve on your own, the solution and answer are at the end of the lesson.

Let's complicate the task a little:

Example 3

Calculate definite integral

Solution:

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

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

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

(especially when the substitution of the upper and lower limits is carried out verbally and is not written out 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 am used to solving such integrals like this:

Here I verbally used the rules of linearity and verbally integrated using the table. I ended up with just one bracket with the limits marked out:

(unlike three brackets in the first method). And into the “whole” antiderivative function, I first substituted 4, then –2, again performing all the actions in my mind.

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

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

is in one bracket.

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

Integrals can be indefinite or definite. Let's consider the form of the definite integral and try to understand its physical meaning. It is represented in this form: $$ \int ^a _b f(x) dx $$. A distinctive feature of writing a definite integral from an indefinite integral is that there are limits of integration a and b. Now we’ll find out why they are needed, and what a definite integral actually 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.

From Fig. 1 it is clear that the definite integral is the same area that is shaded in gray. Let's check this 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 of multiplying the length by the width.

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

The question arises: how to solve indefinite integrals and what is their meaning? Solving such integrals is finding 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 need to independently memorize the properties of integrals and the table of integration of the simplest elementary functions. The 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) $ over a variable. If the function is tabular, then the answer is written in the appropriate form. If not, then the process comes down to obtaining a tabular function from the function $ f(x) $ through tricky mathematical transformations. There are various methods and properties for this, which we will consider further.

So, now let’s create an algorithm for solving integrals for dummies?

Algorithm for calculating integrals

Let's 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 leading to a tabular form of the function $ f(x) $.
If defined, then you need to perform step 2, and then substitute the limits $ a $ and $ b $ into the antiderivative function $ F(x) $. You will find out what formula to use to do this in the article “Newton-Leibniz Formula”.

Examples of solutions

So, you have learned how to solve integrals for dummies, examples of solving integrals have been sorted out. We learned their physical and geometric meaning. The solution methods will be described in other articles.

In each chapter there will be tasks for independent solution, to which you can see the answers.

The concept of a definite integral and the Newton-Leibniz formula

By a definite integral from a continuous function f(x) on the final segment [ a, b] (where ) is called the increment of some of it antiderivative on this segment. (In general, understanding will be much easier if you repeat the topic indefinite integral) In this case, the notation is used

As can be seen in the graphs below (the increment of the antiderivative function is indicated by ), a definite integral can be either a positive or a negative number(It is calculated as the difference between the value of the antiderivative in the upper limit and its value in the lower limit, i.e. as F(b) - F(a)).

Numbers a And b are called the lower and upper limits of integration, respectively, and the segment [ a, b] – segment of integration.

Thus, if F(x) – some antiderivative function for f(x), then, according to the definition,

(38)

Equality (38) is called Newton-Leibniz formula . Difference F(b) – F(a) is briefly written as follows:

Therefore, we will write the Newton-Leibniz formula like this:

(39)

Let us prove that the definite integral does not depend on which antiderivative of the integrand is taken when calculating it. Let F(x) and F( X) are arbitrary antiderivatives of the integrand. Since these are antiderivatives of the same function, they differ by a constant term: Ф( X) = F(x) + C. That's why

This establishes that on the segment [ a, b] increments of all antiderivatives of the function f(x) match up.

Thus, to calculate a definite integral, it is necessary to find any antiderivative of the integrand, i.e. First you need to find the indefinite integral. Constant WITH excluded from subsequent calculations. Then the Newton-Leibniz formula is applied: the value of the upper limit is substituted into the antiderivative function b , further - the value of the lower limit a and the difference is calculated F(b) - F(a) . The resulting number will be a definite integral..

At a = b by definition accepted

Example 1.

Solution. First, let's find the indefinite integral:

Applying the Newton-Leibniz formula to the antiderivative

(at WITH= 0), we get

However, when calculating a definite integral, it is better not to find the antiderivative separately, but to immediately write the integral in the form (39).

Example 2. Calculate definite integral

Solution. Using formula

Find the definite integral yourself and then look at the solution

Properties of the definite integral

Theorem 2.The value of the definite integral does not depend on the designation of the integration variable, i.e.

(40)

Let F(x) – antiderivative for f(x). For f(t) the antiderivative is the same function F(t), in which the independent variable is only designated differently. Hence,

Based on formula (39), the last equality means the equality of the integrals

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

(41)

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

(42)

Theorem 5.If a segment of integration is divided into parts, then the definite integral over the entire segment is equal to the sum of definite integrals over its parts, i.e. If

(43)

Theorem 6.When rearranging the limits of integration, the absolute value of the definite integral does not change, but only its sign changes, i.e.

(44)

Theorem 7(mean value theorem). A definite integral is equal to the product of the length of the integration segment and the value of the integrand at some point inside it, i.e.

(45)

Theorem 8.If the upper limit of integration is greater than the lower one and the integrand is non-negative (positive), then the definite integral is also non-negative (positive), i.e. If

Theorem 9.If the upper limit of integration is greater than the lower one and the functions and are continuous, then the inequality

can be integrated term by term, i.e.

(46)

The properties of the definite integral make it possible to simplify the direct calculation of integrals.

Example 5. Calculate definite integral

Using Theorems 4 and 3, and when finding antiderivatives - table integrals(7) and (6), we get

Definite integral with variable upper limit

Let f(x) – continuous on the segment [ a, b] function, and F(x) is its antiderivative. Consider the definite integral

(47)

and through t the integration variable is designated so as not to confuse it with the upper bound. When it changes X the definite integral (47) also changes, i.e. it is a function of the upper limit of integration X, which we denote by F(X), i.e.

(48)

Let us prove that the function F(X) is an antiderivative for f(x) = f(t). Indeed, differentiating F(X), we get

because F(x) – antiderivative for f(x), A F(a) is a constant value.

Function F(X) – one of the infinite number of antiderivatives for f(x), namely the one that x = a goes to zero. This statement is obtained if in equality (48) we put x = a and use Theorem 1 of the previous paragraph.

Calculation of definite integrals by the method of integration by parts and the method of change of variable

where, by definition, F(x) – antiderivative for f(x). If we change the variable in the integrand

then, in accordance with formula (16), we can write

In this expression

antiderivative function for

In fact, its derivative, according to rule of differentiation of complex functions, is equal

Let α and β be the values of the variable t, for which the function

takes values accordingly a And b, i.e.

But, according to the Newton-Leibniz formula, the difference F(b) – F(a) There is

Definite integral. Examples of solutions

Hello again. In this lesson we will examine in detail such a wonderful thing as a definite integral. This time the introduction will be short. All. Because there is a snowstorm outside the window.

In order to learn how to solve definite integrals you need to:

1) Be able to find indefinite integrals.

2) Be able to calculate definite integral.

In general form, the definite integral is written as follows:

What is added compared to the indefinite integral? More limits of integration.

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

Before we move on to practical examples, a quick 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? Using 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 entire 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 marking. Why is the recording itself needed? Preparation for applying the Newton-Leibniz formula.

2) Substitute the value of the upper limit into the antiderivative function: .

3) 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 segment of integration is not included in the domain of definition 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 points , , and therefore such a definite integral also does not exist. By the way, who hasn’t read the teaching material yet? Graphs and basic properties of elementary functions– the time to do it is now. It will be great to help throughout the course of higher mathematics.

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

???! You cannot substitute negative numbers under the root! What the hell is this?! Initial inattention.

If for a solution (in a test, test, 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 an 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 lower limit of integration be greater than the upper limit of integration? Perhaps this situation actually occurs in practice.

– the integral can be easily calculated using the Newton-Leibniz formula.

What is higher mathematics indispensable? Of course, without all sorts of properties. Therefore, let's consider some properties of the definite integral.

In a definite integral, you can rearrange the upper and lower limits, 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 it is much more convenient to integrate.

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

In a definite integral one can carry out replacement of integration variable, however, compared to the indefinite integral, this has its own specifics, which we will talk about later.

For a definite integral the following holds true: integration by parts formula:

Example 1

Solution:

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

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

Example 2

Calculate definite integral

This is an example for you to solve on your own, the solution and answer are at the end of the lesson.

Let's complicate the task a little:

Example 3

Calculate definite integral

Solution:

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

(2) We integrate according to 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:

THE WEAK LINK in the definite integral is calculation errors and common CONFUSION IN SIGNS. Be careful! I focus special attention on the third term: – first place in the hit parade of errors due to inattention, very often they write automatically (especially when the substitution of the upper and lower limits is carried out verbally and is not written out in such detail). Once again, carefully study the above example.

Here I verbally used the rules of linearity and verbally integrated using the table. I ended up with just one bracket with the limits marked out: (unlike three brackets in the first method). And into the “whole” antiderivative function, I first substituted 4, then –2, again performing all the actions in my mind.

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

Advice: before using the Newton-Leibniz formula, it is useful to check: was the antiderivative itself 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 the draft whether the indefinite integral was found correctly? Let's differentiate:

The original integrand function has been obtained, which means that the indefinite integral has been found correctly. Now we can apply the Newton-Leibniz formula.

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

Example 4

Calculate definite integral

This is an example for you to solve yourself. Try to solve it in a short and detailed way.

Changing a variable in a definite integral

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

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

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

Example 5

Calculate definite integral

The main question here is not the definite integral, but how to correctly carry out the replacement. Let's look at table of integrals and figure out what our integrand function looks like most? Obviously, for the long logarithm: . But there is one discrepancy, in the table integral under the root, and in ours - “x” to the fourth power. The idea of replacement also follows from the reasoning - it would be nice to somehow turn our fourth power into a square. It's real.

First, we prepare our integral for replacement:

From the above considerations, a replacement quite naturally arises:
Thus, everything will be fine in the denominator: .
We find out what the remaining part of the integrand will turn into, for this we find the differential:

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

Finding new limits of integration.

It's quite simple. Let's 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...

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 don’t have to do this) so that it does not interfere with 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, there is no need to carry out any reverse replacements.

And now a couple of examples for you to decide for yourself. What replacements to make - try to guess on your own.

Example 6

Calculate definite integral

Example 7

Calculate definite integral

These are examples for you to decide on your own. 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 site visitors. The first one concerns legality of replacement. In some cases it cannot be done! Thus, Example 6, it would seem, can be solved using universal trigonometric substitution, however, 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 integration segment.

In another email, the following question was received: “Do we need to change the limits of integration when we subsume a function under the differential sign?” At first I wanted to “dismiss the nonsense” and automatically answer “of course not,” but then I thought about the reason for such a question and suddenly discovered that there was no information lacks. But it, although obvious, is very important:

If we subsume the function under the differential sign, 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 summation 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 put the function under the differential sign! It's faster, it's more compact, and it's commonplace - as you'll 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 calculations of the article Integration by parts in the indefinite integral are fully valid for the definite integral.
There is only one detail that is a plus; 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 the example, I again chose the type of integral that has not yet been found anywhere on the site. The example is not the simplest, but very, very informative.

Example 8

Calculate definite integral

Let's decide.

Let's integrate by parts:

Anyone having difficulty with the integral, take a look at the lesson Integrals of trigonometric functions, it is discussed in detail there.

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

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

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

To be honest, I don't like the formula. and, if possible, ... I do without it at all! Let's consider the second solution; from my point of view, it is more rational.

Calculate definite integral

At the first stage I find the indefinite integral:

Let's integrate by parts:

The antiderivative function has been found. There is no point in adding a constant in this case.

What is the advantage of such a hike? There is no need to “carry around” the limits of integration; indeed, it can be exhausting to write down the small symbols of the limits of integration a dozen times

At the second stage I check(usually in draft).

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

The original integrand function 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” solution method there is a much lower risk of getting confused in substitutions and calculations - the Newton-Leibniz formula is applied only once. If the teapot solves a similar integral using the formula (in the first way), then he will definitely make a mistake somewhere.

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

Dear student, print and save:

What to do if you are given a definite integral 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, there is no point in further rocking 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 - this is the weakest link of the task.

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

Example 9

Calculate definite integral

The solution and the answer are somewhere nearby.

The next recommended lesson on the topic is How to calculate the area of a figure using a definite integral?
Let's integrate by parts:

Are you sure you solved them and got these answers? ;-) And there is porn for an old woman.