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 its 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 knowledge of the basics of 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, an integral is a derivative in reverse or an 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 put 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 a figure, the mass of a non-uniform body, the distance traveled during uneven movement, 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 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"
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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 of the indefinite 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 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.
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.
Finding an indefinite integral (a set of antiderivatives or “antiderivatives”) means reconstructing a function from the 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).
Finding an indefinite integral requires a fairly small 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 practicing integration, you need to activate the expression transformation skills you acquired in high school.
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 the algebraic sum of the 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 a 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.
Integral calculus.
Antiderivative function.
Definition: The function F(x) is called antiderivative function function f(x) on the segment if the equality is true at any point of this segment:
It should be noted that there can be an infinite number of antiderivatives for the same function. They will differ from each other by some constant number.
F 1 (x) = F 2 (x) + C.
Indefinite integral.
Definition: Indefinite integral functionf(x) is a set of antiderivative functions that are defined by the relation:
Write down:
The condition for the existence of an indefinite integral on a certain segment is the continuity of the function on this segment.
Properties:
1.
2.
3.
4.
Example:
Finding the value of the indefinite integral is mainly associated with finding the antiderivative of the function. For some functions this is quite a difficult task. Below we will consider methods for finding indefinite integrals for the main classes of functions - rational, irrational, trigonometric, exponential, etc.
For convenience, the values of the indefinite integrals of most elementary functions are collected in special tables of integrals, which are sometimes quite voluminous. They include various commonly used combinations of functions. But most of the formulas presented in these tables are consequences of each other, so below we present a table of basic integrals, with the help of which you can obtain the values of indefinite integrals of various functions.
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Integral |
Meaning |
Integral |
Meaning |
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lnsinx+ C |
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ln |
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Integration methods.
Let's consider three main methods of integration.
Direct integration.
The direct integration method is based on the assumption of the possible value of the antiderivative function with further verification of this value by differentiation. In general, we note that differentiation is a powerful tool for checking the results of integration.
Let's look at the application of this method using an example:
We need to find the value of the integral 
. Based on the well-known differentiation formula 
we can conclude that the sought integral is equal to 
, where C is some constant number. However, on the other hand 
. Thus, we can finally conclude:
Note that, in contrast to differentiation, where clear techniques and methods were used to find the derivative, rules for finding the derivative, and finally the definition of the derivative, such methods are not available for integration. If, when finding the derivative, we used, so to speak, constructive methods, which, based on certain rules, led to the result, then when finding the antiderivative we have to rely mainly on the knowledge of tables of derivatives and antiderivatives.
As for the direct integration method, it is applicable only for some very limited classes of functions. There are very few functions for which you can immediately find an antiderivative. Therefore, in most cases, the methods described below are used.
Method of substitution (replacing variables).
Theorem:
If you need to find the integral 
, but it is difficult to find the antiderivative, then using the replacement x = (t) and dx = (t) dt it turns out:
Proof : Let us differentiate the proposed equality:
According to property No. 2 of the indefinite integral discussed above:
f(x) dx = f[ (t)] (t) dt
which, taking into account the introduced notation, is the initial assumption. The theorem has been proven.
Example. Find the indefinite integral 
.
Let's make a replacement t = sinx, dt = cosxdt.
Example.
Replacement 
We get:
Below we will consider other examples of using the substitution method for various types of functions.
Integration by parts.
The method is based on the well-known formula for the derivative of a product:
(uv)=uv+vu
where uиv are some functions of x.
In differential form: d(uv) =udv+vdu
Integrating, we get: 
, and in accordance with the above properties of the indefinite integral:
or 
;
We have obtained a formula for integration by parts, which allows us to find the integrals of many elementary functions.
Example.
As you can see, consistent application of the integration by parts formula allows you to gradually simplify the function and bring the integral to a tabular one.
Example.
It can be seen that as a result of repeated application of integration by parts, the function could not be simplified to tabular form. However, the last integral obtained is no different from the original one. Therefore, we move it to the left side of the equality.
Thus, the integral was found without using tables of integrals at all.
Before considering in detail the methods of integrating various classes of functions, we give several more examples of finding indefinite integrals by reducing them to tabular ones.
Example.
Example.
Example.
Example.
Example.
Example.
Example.
Example.
Example.
Example.
Integration of elementary fractions.
Definition: Elementary The following four types of fractions are called:
I. 
III. 
II. 
IV. 
m,n – natural numbers (m2,n2) and b 2 – 4ac<0.
The first two types of integrals of elementary fractions can be quite simply brought to tables by substituting t=ax+b.
Let us consider the method of integrating elementary fractions of type III.
The integral of a fraction of type III can be represented as:
Here, in general form, the reduction of a fraction integral of type III to two tabular integrals is shown.
Let's look at the application of the above formula using examples.
Example.
Generally speaking, if the trinomial ax 2 +bx+c has the expression b 2 – 4ac>0, then the fraction, by definition, is not elementary, however, it can nevertheless be integrated in the manner indicated above.
Example.
Example.
Let us now consider methods for integrating simple fractions of type IV.
First, let's consider a special case with M = 0, N = 1.
Then the integral of the form 
can be represented in the form by selecting the complete square in the denominator 
. Let's make the following transformation:
We will take the second integral included in this equality by parts.
Let's denote: 
For the original integral we obtain:
The resulting formula is called recurrent. If you apply it n-1 times, you get a table integral 
.
Let us now return to the integral of an elementary fraction of type IV in the general case.
In the resulting equality, the first integral using the substitution t
=
u 2
+
s reduced to tabular 
, and the recurrence formula discussed above is applied to the second integral.
Despite the apparent complexity of integrating an elementary fraction of type IV, in practice it is quite easy to use for fractions with a small degree n, and the universality and generality of the approach makes possible a very simple implementation of this method on a computer.
Example:
Integration of rational functions.
Integrating rational fractions.
In order to integrate a rational fraction, it is necessary to decompose it into elementary fractions.
Theorem:
If 
- a proper rational fraction, the denominator P(x) of which is represented as a product of linear and quadratic factors (note that any polynomial with real coefficients can be represented in this form: P(x)
= (x -
a)
…(x
-
b)
(x 2
+
px +
q)
…(x 2
+
rx +
s)
), then this fraction can be decomposed into elementary ones according to the following scheme:
where A i ,B i ,M i ,N i ,R i ,S i are some constant quantities.
When integrating rational fractions, they resort to decomposing the original fraction into elementary ones. To find the quantities A i , B i , M i , N i , R i , S i the so-called method of uncertain coefficients, the essence of which is that in order for two polynomials to be identically equal, it is necessary and sufficient that the coefficients at the same powers of x be equal.
Let's consider the use of this method using a specific example.
Example.
Reducing to a common denominator and equating the corresponding numerators, we get:
Example.
Because If the fraction is improper, you must first select its whole part:
6
x 5 – 8x 4 – 25x 3 + 20x 2 – 76x– 7 3x 3 – 4x 2 – 17x+ 6
6x 5 – 8x 4 – 34x 3 + 12x 2 2x 2 + 3
9x 3 + 8x 2 – 76x - 7
9x 3 – 12x 2 – 51x +18
20x 2 – 25x – 25
Let's factorize the denominator of the resulting fraction. It can be seen that at x = 3 the denominator of the fraction turns to zero. Then:
3x 3 – 4x 2 – 17x+ 6x- 3
3x 3 – 9x 2 3x 2 + 5x- 2
Thus 3x 3 – 4x 2 – 17x+ 6 = (x– 3)(3x 2 + 5x– 2) = (x– 3)(x+ 2)(3x– 1). Then:
In order to avoid opening brackets, grouping and solving a system of equations (which in some cases may turn out to be quite large) when finding uncertain coefficients, the so-called arbitrary value method. The essence of the method is that several (according to the number of undetermined coefficients) arbitrary values of x are substituted into the above expression. To simplify calculations, it is customary to take as arbitrary values points at which the denominator of the fraction is equal to zero, i.e. in our case – 3, -2, 1/3. We get:
Finally we get:
=
Example.
Let's find the undetermined coefficients:
Then the value of the given integral:
Integration of some trigonometrics
functions.
There can be an infinite number of integrals from trigonometric functions. Most of these integrals cannot be calculated analytically at all, so we will consider some of the most important types of functions that can always be integrated.
Integral of the form 
.
Here R is the designation of some rational function of the variables sinx and cosx.
Integrals of this type are calculated using substitution 
. This substitution allows you to convert a trigonometric function to a rational one.
,
Then 
Thus:
The transformation described above is called universal trigonometric substitution.
Example.
The undoubted advantage of this substitution is that with its help you can always transform a trigonometric function into a rational one and calculate the corresponding integral. The disadvantages include the fact that the transformation can result in a rather complex rational function, the integration of which will take a lot of time and effort.
However, if it is impossible to apply a more rational replacement of the variable, this method is the only effective one.
Example.
Integral of the form 
If
functionRcosx.
Despite the possibility of calculating such an integral using the universal trigonometric substitution, it is more rational to use the substitution t = sinx.
Function 
can contain cosx only in even powers, and therefore can be converted into a rational function with respect to sinx.
Example.
Generally speaking, to apply this method, only the oddness of the function relative to the cosine is necessary, and the degree of the sine included in the function can be any, both integer and fractional.
Integral of the form 
If
functionRis odd relative tosinx.
By analogy with the case considered above, the substitution is made t = cosx.
Example.
Integral of the form 
functionReven relativelysinxAndcosx.
To transform the function R into a rational one, use the substitution
t = tgx.
Example.
Integral of the product of sines and cosines
various arguments.
Depending on the type of work, one of three formulas will be applied:
Example.
Example.
Sometimes when integrating trigonometric functions it is convenient to use well-known trigonometric formulas to reduce the order of functions.
Example.
Example.
Sometimes some non-standard techniques are used.
Example.
Integration of some irrational functions.
Not every irrational function can have an integral expressed by elementary functions. To find the integral of an irrational function, you should use a substitution that will allow you to transform the function into a rational one, the integral of which can always be found, as is always known.
Let's look at some techniques for integrating various types of irrational functions.
Integral of the form 
Wheren- natural number.
Using substitution 
the function is rationalized.
Example.
If the composition of an irrational function includes roots of various degrees, then as a new variable it is rational to take the root of a degree equal to the least common multiple of the degrees of the roots included in the expression.
Let's illustrate this with an example.
Example.
Integration of binomial differentials.
Definition: Binomial differential called expression
x m (a + bx n ) p dx
Where m, n, And p– rational numbers.
As was proven by academician P.L. Chebyshev. (1821-1894), the integral of a binomial differential can be expressed in terms of elementary functions only in the following three cases:
If R is an integer, then the integral is rationalized using the substitution
, where is the common denominator m And n.
