At school, many people fail to solve integrals or have any difficulties with them. This article will help you figure it out, as you will find everything in it. integral tables.
Integral is one of the main calculations and concepts in mathematical analysis. Its appearance resulted from two purposes:
First goal- restore a function using its derivative.
Second goal- calculation of the area located at the distance from the graph to the function f(x) on the straight line where, a is greater than or equal to x greater than or equal to b and the x-axis.
These goals lead us to definite and indefinite integrals. The connection between these integrals lies in the search for properties and calculation. But everything flows and everything changes over time, new solutions were found, additions were identified, thereby leading definite and indefinite integrals to other forms of integration.
What's happened indefinite integral you ask. This is an antiderivative function F(x) of one variable x in the interval a greater than x greater than b. is called any function F(x), in a given interval for any designation x, the derivative is equal to F(x). It is clear that F(x) is antiderivative for f(x) in the interval a is greater than x is greater than b. This means F1(x) = F(x) + C. C - is any constant and antiderivative for f(x) in a given interval. This statement is invertible; for the function f(x) - 2 the antiderivatives differ only in the constant. Based on the theorem integral calculus it turns out that each continuous in the interval a
Definite integral is understood as a limit in integral sums, or in the situation of a given function f(x) defined on some line (a,b) having an antiderivative F on it, meaning the difference of its expressions at the ends of a given line F(b) - F(a).
To illustrate the study of this topic, I suggest watching the video. It tells in detail and shows how to find integrals.
Each table of integrals in itself is very useful, as it helps in solving specific type integrals.
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Let us list the integrals of elementary functions, which are sometimes called tabular:
Any of the above formulas can be proven by taking the derivative of the right-hand side (the result will be the integrand).
Integration methods
Let's look at some basic integration methods. These include:
1. Decomposition method(direct integration).
This method is based on the direct use of tabular integrals, as well as on the use of properties 4 and 5 of the indefinite integral (i.e., taking the constant factor out of brackets and/or representing the integrand as a sum of functions - decomposition of the integrand into terms).
Example 1. For example, to find(dx/x 4) you can directly use the table integral forx n dx. In fact,(dx/x 4) =x -4 dx=x -3 /(-3) +C= -1/3x 3 +C.
Let's look at a few more examples.
Example 2. To find it, we use the same integral:
Example 3. To find it you need to take
Example 4. To find, we represent the integrand function in the form 
and use the table integral for the exponential function:
Let's consider the use of bracketing a constant factor.
Example 5.
Let's find, for example 
. Considering that, we get
Example 6. We'll find it. Because the 
, let's use the table integral 
We get
In the following two examples, you can also use bracketing and table integrals:
Example 7.
(we use and 
);
Example 8.
(we use 
And 
).
Let's look at more complex examples that use the sum integral.
Example 9. For example, let's find 
. To apply the expansion method in the numerator, we use the sum cube formula , and then divide the resulting polynomial by the denominator, term by term.
=((8x 3/2 + 12x+ 6x 1/2 + 1)/(x 3/2))dx=(8 + 12x -1/2 + 6/x+x -3/2)dx= 8 dx+ 12x -1/2 dx+ + 6dx/x+x -3/2 dx= 
It should be noted that at the end of the solution one common constant C is written (and not separate ones when integrating each term). In the future, it is also proposed to omit the constants from the integration of individual terms in the solution process as long as the expression contains at least one indefinite integral (we will write one constant at the end of the solution).
Example 10. We'll find 
. To solve this problem, let's factorize the numerator (after this we can reduce the denominator).
Example 11. We'll find it. Trigonometric identities can be used here.
Sometimes, in order to decompose an expression into terms, you have to use more complex techniques.
Example 12. We'll find 
. In the integrand we select the whole part of the fraction 
. Then
Example 13. We'll find
2. Variable replacement method (substitution method)
The method is based on the following formula: f(x)dx=f((t))`(t)dt, where x =(t) is a function differentiable on the interval under consideration.
Proof. Let us find the derivatives with respect to the variable tfrom the left and right parts formulas.
Note that on the left side there is a complex function whose intermediate argument is x = (t). Therefore, to differentiate it with respect to t, we first differentiate the integral with respect to x, and then take the derivative of the intermediate argument with respect to t.
( f(x)dx)` t = ( f(x)dx)` x *x` t = f(x) `(t)
Derivative from the right side:
(f((t))`(t)dt)` t =f((t))`(t) =f(x)`(t)
Since these derivatives are equal, by corollary to Lagrange’s theorem, the left and right sides of the formula being proved differ by a certain constant. Since the indefinite integrals themselves are defined up to an indefinite constant term, this constant can be omitted from the final notation. Proven.
A successful change of variable allows you to simplify the original integral, and in the simplest cases, reduce it to a tabular one. In the application of this method, a distinction is made between linear and nonlinear substitution methods.
a) Linear substitution method Let's look at an example.
Example 1.
. Let t= 1 – 2x, then
dx=d(½ - ½t) = - ½dt
It should be noted that the new variable does not need to be written out explicitly. In such cases, they talk about transforming a function under the differential sign or about introducing constants and variables under the differential sign, i.e. O implicit variable replacement.
Example 2. For example, let's findcos(3x + 2)dx. By the properties of the differential dx = (1/3)d(3x) = (1/3)d(3x + 2), thencos(3x + 2)dx =(1/3)cos(3x + 2)d (3x + + 2) = (1/3)cos(3x + 2)d(3x + 2) = (1/3)sin(3x + 2) +C.
In both examples considered, linear substitution t=kx+b(k0) was used to find the integrals.
In the general case, the following theorem is valid.
Linear substitution theorem. Let F(x) be some antiderivative of the function f(x). Thenf(kx+b)dx= (1/k)F(kx+b) +C, where k and b are some constants,k0.
Proof.
By definition of the integral f(kx+b)d(kx+b) =F(kx+b) +C. Hod(kx+b)= (kx+b)`dx=kdx. Let's take the constant factor k out of the integral sign: kf(kx+b)dx=F(kx+b) +C. Now we can divide the left and right sides of the equality into two and obtain the statement to be proved up to the designation of the constant term.
This theorem states that if in the definition of the integral f(x)dx= F(x) + C instead of the argument x we substitute the expression (kx+b), this will lead to the appearance of an additional factor 1/k in front of the antiderivative.
Using the proven theorem, we solve the following examples.
Example 3.
We'll find 
. Here kx+b= 3 –x, i.e. k= -1,b= 3. Then
Example 4.
We'll find it. Herekx+b= 4x+ 3, i.e. k= 4,b= 3. Then
Example 5.
We'll find 
. Here kx+b= -2x+ 7, i.e. k= -2,b= 7. Then
.
Example 6. We'll find 
. Here kx+b= 2x+ 0, i.e. k= 2,b= 0.
.
Let us compare the result obtained with example 8, which was solved by the decomposition method. Solving the same problem using a different method, we got the answer 
. Let's compare the results: Thus, these expressions differ from each other by a constant term 
, i.e. The answers received do not contradict each other.
Example 7. We'll find 
. Let's select a perfect square in the denominator.
In some cases, changing a variable does not reduce the integral directly to a tabular one, but can simplify the solution, making it possible to use the expansion method at a subsequent step.
Example 8. For example, let's find 
. Replace t=x+ 2, then dt=d(x+ 2) =dx. Then
,
where C = C 1 – 6 (when substituting the expression (x+ 2) instead of the first two terms we get ½x 2 -2x– 6).
Example 9. We'll find 
. Let t= 2x+ 1, then dt= 2dx;dx= ½dt;x= (t– 1)/2.
Let's substitute the expression (2x+ 1) for t, open the brackets and give similar ones.
Note that in the process of transformations we moved to another constant term, because the group of constant terms could be omitted during the transformation process.
b) Nonlinear substitution method Let's look at an example.
Example 1.
. Lett= -x 2. Next, one could express x in terms of t, then find an expression for dx and implement a change of variable in the desired integral. But in this case it’s easier to do things differently. Let's finddt=d(-x 2) = -2xdx. Note that the expression xdx is a factor of the integrand of the desired integral. Let us express it from the resulting equalityxdx= - ½dt. Then
Antiderivative function and indefinite integral
Fact 1. Integration is the inverse action of differentiation, namely, restoring a function from the known derivative of this function. The function thus restored F(x) is called antiderivative for function f(x).
Definition 1. Function F(x f(x) on some interval X, if for all values x from this interval the equality holds F "(x)=f(x), that is this function f(x) is a derivative of antiderivative function F(x). .
For example, the function F(x) = sin x is an antiderivative of the function f(x) = cos x on the entire number line, since for any value of x (sin x)" = (cos x) .
Definition 2. Indefinite integral of a function f(x) is the set of all its antiderivatives. In this case, the notation is used
∫
f(x)dx
,where is the sign ∫ called the integral sign, the function f(x) – integrand function, and f(x)dx – integrand expression.
Thus, if F(x) – some antiderivative for f(x) , That
∫
f(x)dx = F(x) +C
Where C - arbitrary constant (constant).
To understand the meaning of the set of antiderivatives of the function as indefinite integral The following analogy is appropriate. Let there be a door (traditional wooden door). Its function is to “be a door.” What is the door made of? Made of wood. This means that the set of antiderivatives of the integrand of the function “to be a door”, that is, its indefinite integral, is the function “to be a tree + C”, where C is a constant that in in this context may indicate, for example, a type of tree. Just as a door is made from wood using some tools, a derivative of a function is “made” from an antiderivative function using formulas we learned while studying the derivative .
Then the table of functions of common objects and their corresponding antiderivatives (“to be a door” - “to be a tree”, “to be a spoon” - “to be metal”, etc.) is similar to the table of basic indefinite integrals, which will be given below. The table of indefinite integrals lists common functions with an indication of the antiderivatives from which these functions are “made”. In part of the problems on finding the indefinite integral, integrands are given that can be integrated directly without much effort, that is, using the table of indefinite integrals. In more complex problems, the integrand must first be transformed so that table integrals can be used.
Fact 2. When restoring a function as an antiderivative, we must take into account an arbitrary constant (constant) C, and in order not to write a list of antiderivatives with various constants from 1 to infinity, you need to write a set of antiderivatives with an arbitrary constant C, for example, like this: 5 x³+C. So, an arbitrary constant (constant) is included in the expression of the antiderivative, since the antiderivative can be a function, for example, 5 x³+4 or 5 x³+3 and when differentiated, 4 or 3, or any other constant goes to zero.
Let us pose the integration problem: for this function f(x) find such a function F(x), whose derivative equal to f(x).
Example 1. Find the set of antiderivatives of a function
Solution. For this function, the antiderivative is the function
Function F(x) is called an antiderivative for the function f(x), if the derivative F(x) is equal to f(x), or, which is the same thing, differential F(x) is equal f(x) dx, i.e.
(2)
Therefore, the function is an antiderivative of the function. However, it is not the only antiderivative for . They also serve as functions
Where WITH– arbitrary constant. This can be verified by differentiation.
Thus, if there is one antiderivative for a function, then for it there is infinite set antiderivatives that differ by a constant term. All antiderivatives for a function are written in the above form. This follows from the following theorem.
Theorem (formal statement of fact 2). If F(x) – antiderivative for the function f(x) on some interval X, then any other antiderivative for f(x) on the same interval can be represented in the form F(x) + C, Where WITH– arbitrary constant.
IN following example We already turn to the table of integrals, which will be given in paragraph 3, after the properties of the indefinite integral. We do this before reading the entire table so that the essence of the above is clear. And after the table and properties, we will use them in their entirety during integration.
Example 2. Find sets of antiderivative functions:
Solution. We find sets of antiderivative functions from which these functions are “made”. When mentioning formulas from the table of integrals, for now just accept that there are such formulas there, and we will study the table of indefinite integrals itself a little further.
1) Applying formula (7) from the table of integrals for n= 3, we get
2) Using formula (10) from the table of integrals for n= 1/3, we have
3) Since
then according to formula (7) with n= -1/4 we find
It is not the function itself that is written under the integral sign. f, and its product by the differential dx. This is done primarily in order to indicate by which variable the antiderivative is sought. For example,
,
;
here in both cases the integrand is equal to , but its indefinite integrals in the cases considered turn out to be different. In the first case, this function is considered as a function of the variable x, and in the second - as a function of z .
The process of finding the indefinite integral of a function is called integrating that function.
Geometric meaning of the indefinite integral
Suppose we need to find a curve y=F(x) and we already know that the tangent of the angle of inclination of the tangent at each point is given function f(x) abscissa of this point.
According to geometric sense derivative, tangent of the tangent angle at a given point on the curve y=F(x) equal to the value derivative F"(x). So we need to find such a function F(x), for which F"(x)=f(x). Function required in the task F(x) is an antiderivative of f(x). The conditions of the problem are satisfied not by one curve, but by a family of curves. y=F(x)- one of these curves, and any other curve can be obtained from it parallel transfer along the axis Oy.
Let's call the graph of the antiderivative function of f(x) integral curve. If F"(x)=f(x), then the graph of the function y=F(x) there is an integral curve.
Fact 3. The indefinite integral is geometrically represented by the family of all integral curves , as in the picture below. The distance of each curve from the origin of coordinates is determined by an arbitrary integration constant C.
Properties of the indefinite integral
Fact 4. Theorem 1. The derivative of an indefinite integral is equal to the integrand, and its differential is equal to the integrand.
Fact 5. Theorem 2. Indefinite integral of the differential of a function f(x) equal to function f(x) up to a constant term , i.e.
(3)
Theorems 1 and 2 show that differentiation and integration are mutually inverse operations.
Fact 6. Theorem 3. Constant multiplier V integrand can be taken out of the sign of the indefinite integral , i.e.
