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 the simplest and other integrals and why you can’t do without it in mathematics.
We study the concept « 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 a basic understanding of the basics. mathematical analysis. We already have information about limits and derivatives, necessary for understanding integrals, 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, an integral is a derivative in reverse or an antiderivative. By the way, read our article about how to calculate derivatives.
The antiderivative exists for everyone 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 elementary functions, it is convenient to summarize them in a table and use ready-made values.
Complete table of integrals for students
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 bounded by the graph of a function? Using an integral! Let's break it down curved trapezoid, limited by the coordinate axes and the graph of the function, into infinitely small 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.
« Integral »
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Rules for calculating integrals for dummies
Properties of the indefinite integral
How to solve an indefinite integral? Here we will look at the properties definite integral, which will be useful when solving examples.
- The derivative of the integral is equal to the integrand:
- The constant can be taken out from under the integral sign:
- The integral of the sum is equal to the sum of the 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 the indefinite integral and examples with solutions. We suggest you 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. Contact a professional student service, and any triple or line integral on a closed surface you will be able to do it.
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 view 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 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, since the segment of integration is not included in the domain of definition of the integrand (the values under 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 it yet? methodological material 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 to solve (in test work, on a 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 the definite integral be equal to 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.
(3) For each of the three terms we apply the Newton-Leibniz formula: 
THE WEAK LINK in the definite integral is calculation errors and the 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.
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 - common fractions I count 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.
>> >> >> Integration methods
Basic integration methods
Definition of integral, definite and indefinite, table of integrals, Newton-Leibniz formula, integration by parts, examples of calculating integrals.
Indefinite integral
Let u = f(x) and v = g(x) be functions that have continuous . Then, according to the work,
d(uv))= udv + vdu or udv = d(uv) - vdu.
For the expression d(uv), the antiderivative will obviously be uv, so the formula holds:
∫ udv = uv - ∫ vdu (8.4.)
This formula expresses the rule integration by parts. It leads the integration of the expression udv=uv"dx to the integration of the expression vdu=vu"dx.
Let, for example, you want to find ∫xcosx dx. Let us put u = x, dv = cosxdx, so du=dx, v=sinx. Then
∫xcosxdx = ∫x d(sin x) = x sin x - ∫sin x dx = x sin x + cosx + C.
The rule of integration by parts has a more limited scope than substitution of variables. But there are whole classes of integrals, for example, ∫x k ln m xdx, ∫x k sinbxdx, ∫ x k cosbxdx, ∫x k e ax and others, which are calculated precisely using integration by parts.
Definite integral
Integration methods, the concept of a definite integral is introduced as follows. Let a function f(x) be defined on an interval. Let us divide the segment [a,b] into n parts by points a= x 0< x 1 <...< x n = b. Из каждого интервала (x i-1 ,
x i) возьмем произвольную точку ξ i и составим сумму f(ξ i)
Δx i где
Δ x i =x i - x i-1. A sum of the form f(ξ i)Δ x i is called an integral sum, and its limit at λ = maxΔx i → 0, if it exists and is finite, is called definite integral functions f(x) from a to b and is denoted:
F(ξ i)Δx i (8.5).
The function f(x) in this case is called integrable on the interval, numbers a and b are called lower and upper limits of the integral.
Integration methods have the following properties:
The last property is called mean value theorem.
Let f(x) be continuous on . Then on this segment there is an indefinite integral
∫f(x)dx = F(x) + C
and takes place Newton-Leibniz formula, connecting the definite integral with the indefinite integral:
F(b) - F(a). (8.6)
Geometric interpretation: represents the area of a curvilinear trapezoid bounded from above by the curve y=f(x), straight lines x = a and x = b and a segment of the Ox axis.
Improper integrals
Integrals with infinite limits and integrals of discontinuous (unbounded) functions are called improper. Improper integrals of the first kind - These are integrals over an infinite interval, defined as follows:
(8.7)
If this limit exists and is finite, then it is called a convergent improper integral of f(x) on the interval [a,+ ∞), and the function f(x) is called integrable on the infinite interval [a,+ ∞). Otherwise, the integral is said to not exist or to diverge.
Improper integrals on the intervals (-∞,b] and (-∞, + ∞) are defined similarly:
Let us define the concept of an integral of an unbounded function. If f(x) is continuous for all values x of the segment except for the point c, at which f(x) has an infinite discontinuity, then improper integral of the second kind of f(x) ranging from a to b the amount is called:
if these limits exist and are finite. Designation:
Examples of integral calculations
Example 3.30. Calculate ∫dx/(x+2).
Solution. Let us denote t = x+2, then dx = dt, ∫dx/(x+2) = ∫dt/t = ln|t| + C = ln|x+2| +C.
Example 3.31. Find ∫ tgxdx.
Solution: ∫ tgxdx = ∫sinx/cosxdx = - ∫dcosx/cosx. Let t=cosx, then ∫ tgxdx = -∫ dt/t = - ln|t| + C = -ln|cosx|+C.
Example3.32 . Find ∫dx/sinx
Example3.33. Find .
Solution. =
.
Example3.34 . Find ∫arctgxdx.
Solution. Let's integrate by parts. Let us denote u=arctgx, dv=dx. Then du = dx/(x 2 +1), v=x, whence ∫arctgxdx = xarctgx - ∫ xdx/(x 2 +1) = xarctgx + 1/2 ln(x 2 +1) +C; because
∫xdx/(x 2 +1) = 1/2 ∫d(x 2 +1)/(x 2 +1) = 1/2 ln(x 2 +1) +C.
Example3.35 . Calculate ∫lnxdx.
Solution. Applying the integration by parts formula, we obtain:
u=lnx, dv=dx, du=1/x dx, v=x. Then ∫lnxdx = xlnx - ∫x 1/x dx =
= xlnx - ∫dx + C= xlnx - x + C.
Example3.36 . Calculate ∫e x sinxdx.
Solution. Let's apply the integration by parts formula. Let us denote u = e x, dv = sinxdx, then du = e x dx, v =∫ sinxdx= - cosx → ∫ e x sinxdx = - e x cosx + ∫ e x cosxdx. ∫e x cosxdx also integrate by parts: u = e x , dv = cosxdx, du=e x dx, v=sinx. We have:
∫ e x cosxdx = e x sinx - ∫ e x sinxdx. We obtained the relation ∫e x sinxdx = - e x cosx + e x sinx - ∫ e x sinxdx, from which 2∫e x sinx dx = - e x cosx + e x sinx + C.
Example 3.37. Calculate J = ∫cos(lnx)dx/x.
Solution: Since dx/x = dlnx, then J= ∫cos(lnx)d(lnx). Replacing lnx through t, we arrive at the table integral J = ∫ costdt = sint + C = sin(lnx) + C.
Example 3.38 . Calculate J = .
Solution. Considering that = d(lnx), we substitute lnx = t. Then J = 
.
Example 3.39
. Calculate J = 
.
Solution. We have: 
. That's why =
What are integrals for? Try to answer this question for yourself.
When explaining the topic of integrals, teachers list areas of application that are of little use to school minds. Among them:
- calculating the area of a figure.
- Calculation of body mass with uneven density.
- determining the distance traveled when moving at a variable speed.
- etc.
It is not always possible to connect all these processes, so many students get confused, even if they have all the basic knowledge to understand the integral.
The main reason for ignorance– lack of understanding of the practical significance of integrals.
Integral - what is it?
Prerequisites. The need for integration arose in Ancient Greece. At that time, Archimedes began to use methods that were essentially similar to modern integral calculus to find the area of a circle. The main approach for determining the area of uneven figures then was the “Exhaustion Method”, which is quite easy to understand.
The essence of the method. A monotonic sequence of other figures fits into this figure, and then the limit of the sequence of their areas is calculated. This limit was taken as the area of this figure.
This method easily traces the idea of integral calculus, which is to find the limit of an infinite sum. This idea was later used by scientists to solve applied problems astronautics, economics, mechanics, etc.
Modern integral. The classical theory of integration was formulated in general form by Newton and Leibniz. It relied on the then existing laws of differential calculus. To understand it, you need to have some basic knowledge that will help you use mathematical language to describe visual and intuitive ideas about integrals.
We explain the concept of “Integral”
The process of finding the derivative is called differentiation, and finding the antiderivative – integration.
Integral mathematical language– this is the antiderivative of the function (what was before the derivative) + constant “C”.
Integral in simple words is the area of a curvilinear figure. The indefinite integral is the entire area. The definite integral is the area in a given area.
The integral is written like this:
Each integrand is multiplied by the "dx" component. It shows over which variable the integration is being carried out. "dx" is the increment of the argument. Instead of X there can be any other argument, for example t (time).
Indefinite integral
An indefinite integral has no limits of integration.
To solve indefinite integrals, it is enough to find the antiderivative of the integrand and add “C” to it.
Definite integral
In a definite integral, the restrictions “a” and “b” are written on the integration sign. These are indicated on the X-axis in the graph below.
To calculate a definite integral, you need to find the antiderivative, substitute the values “a” and “b” into it and find the difference. In mathematics this is called Newton-Leibniz formula:
Table of integrals for students (basic formulas)
Download the integral formulas, they will be useful to you
How to calculate the integral correctly
There are several simple operations for transforming integrals. Here are the main ones:
Removing a constant from under the integral sign
Decomposition of the integral of a sum into the sum of integrals
If you swap a and b, the sign will change
You can split the integral into intervals as follows
These are the simplest properties, on the basis of which more complex theorems and methods of calculus will later be formulated.
Examples of integral calculations
Solving the indefinite integral
Solving the definite integral
Basic concepts for understanding the topic
So that you understand the essence of integration and do not close the page from misunderstanding, we will explain a number of basic concepts. What is a function, derivative, limit and antiderivative.
Function– a rule according to which all elements from one set are correlated with all elements from another.
Derivative– a function that describes the rate of change of another function at each specific point. In strict language, this is the limit of the ratio of the increment of a function to the increment of the argument. It is calculated manually, but it is easier to use a derivative table, which contains most of the standard functions.
Increment– a quantitative change in the function with some change in the argument.
Limit– the value to which the function value tends when the argument tends to a certain value.
An example of a limit: let's say if X is equal to 1, Y will be equal to 2. But what if X is not equal to 1, but tends to 1, that is, it never reaches it? In this case, y will never reach 2, but will only tend to this value. In mathematical language this is written as follows: limY(X), as X –> 1 = 2. It reads: the limit of the function Y(X), as x tends to 1, is equal to 2.
As already mentioned, a derivative is a function that describes another function. The original function may be a derivative of some other function. This other function is called antiderivative.
Conclusion
Finding the integrals is not difficult. If you don't understand how to do this, . The second time it becomes clearer. Remember! Solving integrals comes down to simple transformations of the integrand and searching for it in .
If the text explanation doesn’t suit you, watch the video about the meaning of the integral and derivative:
Integrals - what they are, how to solve, examples of solutions and explanation for dummies updated: November 22, 2019 by: Scientific Articles.Ru
The process of solving integrals in the science called mathematics is called integration. Using integration we 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 a definite integral and try to understand it physical meaning. It is represented in this form: $$ \int ^a _b f(x) dx $$. Distinctive feature writing a definite integral of an indefinite integral is that there are limits of integration of a and b. Now we’ll find out why they are needed, and what a definite integral actually means. IN geometric sense such an integral equal to area a 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 fits 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. Finding it 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 table function from the function $ f(x) $ through tricky mathematical transformations. For this there is various methods and properties that 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.
