These properties are used to carry out transformations of the integral with the aim of reducing it to one of the elementary integrals and further calculation.

1. The derivative of the indefinite integral is equal to the integrand:

2. The differential of the indefinite integral is equal to the integrand:

3. The indefinite integral of the differential of some function is equal to the sum of this function and an arbitrary constant:

4. The constant factor can be taken out of the integral sign:

Moreover, a ≠ 0

5. The integral of the sum (difference) is equal to the sum (difference) of the integrals:

6. The property is a combination of properties 4 and 5:

Moreover, a ≠ 0 ˄ b ≠ 0

7. Property of invariance of the indefinite integral:

If, then

8. Property:

If, then

In fact, this property is a special case of integration using the variable change method, which is discussed in more detail in the next section.

Let's consider an example:

First we applied property 5, then property 4, then we used the antiderivatives table and got the result.

The algorithm of our online integral calculator supports all the properties listed above and can easily find a detailed solution for your integral.

Let the function y = f(x) is defined on the segment [ a, b ], a < b... Let's perform the following operations:

1) we split [ a, b] dots a = x 0 < x 1 < ... < x i- 1 < x i < ... < x n = b on n partial line segments [ x 0 , x 1 ], [x 1 , x 2 ], ..., [x i- 1 , x i ], ..., [x n- 1 , x n ];

2) in each of the partial segments [ x i- 1 , x i ], i = 1, 2, ... n, choose an arbitrary point and calculate the value of the function at this point: f(z i ) ;

3) find works f(z i ) · Δ x i , where is the length of the partial segment [ x i- 1 , x i ], i = 1, 2, ... n;

4) compose integral sum function y = f(x) on the segment [ a, b ]:

From a geometric point of view, this sum σ is the sum of the areas of rectangles, the bases of which are partial segments [ x 0 , x 1 ], [x 1 , x 2 ], ..., [x i- 1 , x i ], ..., [x n- 1 , x n ], and the heights are f(z 1 ) , f(z 2 ), ..., f(z n) respectively (Fig. 1). Let us denote by λ length of the largest partial segment:

5) find the limit of the integral sum when λ → 0.

Definition. If there is a finite limit of the integral sum (1) and it does not depend on the method of partitioning the segment [ a, b] to partial segments, nor from the selection of points z i in them, then this limit is called definite integral from function y = f(x) on the segment [ a, b] and is denoted

Thus,

In this case, the function f(x) is called integrable on [ a, b]. The numbers a and b are called, respectively, the lower and upper limits of integration, f(x) Is the integrand, f(x ) dx- the integrand, x- variable of integration; section [ a, b] is called the integration interval.

Theorem 1. If the function y = f(x) is continuous on the segment [ a, b], then it is integrable on this segment.

The definite integral with the same limits of integration is equal to zero:

If a > b, then, by definition, we put

2. Geometric meaning of a definite integral

Let on the segment [ a, b] a continuous nonnegative function is given y = f(x ) . Curved trapezoid is the figure bounded from above by the graph of the function y = f(x), from below - by the Ox axis, to the left and right - by straight lines x = a and x = b(fig. 2).

The definite integral of a non-negative function y = f(x) from a geometric point of view is equal to the area of ​​a curvilinear trapezoid bounded from above by the graph of the function y = f(x), to the left and to the right - by line segments x = a and x = b, below - by a segment of the Ox axis.

3. Basic properties of a definite integral

1. The value of the definite integral does not depend on the designation of the variable of integration:

2. A constant factor can be taken outside the sign of a definite integral:

3. A definite integral of the algebraic sum of two functions is equal to the algebraic sum of definite integrals of these functions:

4.If the function y = f(x) is integrable on [ a, b] and a < b < c, then

5. (mean value theorem)... If the function y = f(x) is continuous on the segment [ a, b], then on this segment there is a point such that

4. Newton-Leibniz formula

Theorem 2. If the function y = f(x) is continuous on the segment [ a, b] and F(x) Is any of its antiderivatives on this segment, then the following formula is valid:

which is called by the Newton – Leibniz formula. Difference F(b) - F(a) it is customary to write as follows:

where the character is called a double wildcard character.

Thus, formula (2) can be written as:

Example 1. Calculate the integral

Solution. For the integrand f(x ) = x 2 an arbitrary antiderivative has the form

Since any antiderivative can be used in the Newton-Leibniz formula, to calculate the integral we take the antiderivative, which has the simplest form:

5. Change of variable in a definite integral

Theorem 3. Let the function y = f(x) is continuous on the segment [ a, b]. If:

1) function x = φ ( t) and its derivative φ "( t) are continuous at;

2) the set of values ​​of the function x = φ ( t) for is the segment [ a, b ];

3) φ ( a) = a, φ ( b) = b, then the formula is valid

which is called by the variable change formula in the definite integral .

Unlike the indefinite integral, in this case not necessary return to the original variable of integration - it is enough just to find new limits of integration α and β (for this it is necessary to solve with respect to the variable t the equation φ ( t) = a and φ ( t) = b).

Instead of substitution x = φ ( t) you can use the substitution t = g(x). In this case, finding new limits of integration with respect to the variable t simplified: α = g(a) , β = g(b) .

Example 2... Calculate the integral

Solution. Let's introduce a new variable by the formula. Squaring both sides of the equality, we get 1 + x = t 2 , where x = t 2 - 1, dx = (t 2 - 1)"dt= 2tdt... We find new limits of integration. To do this, we substitute the old limits into the formula x = 3 and x = 8. We get:, whence t= 2 and α = 2; , where t= 3 and β = 3. So,

Example 3. Calculate

Solution. Let be u= ln x, then , v = x... According to the formula (4)

Antiderivative function and indefinite integral

Fact 1. Integration is an action inverse to differentiation, namely, the restoration of a function from a 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 F "(x)=f(x), that is, this function f(x) is the derivative of the antiderivative function F(x). .

For example, the function F(x) = sin x is the antiderivative of the function f(x) = cos x on the whole number line, since for any value of x (sin x) "= (cos x) .

Definition 2. The indefinite integral of a function f(x) is the set of all its antiderivatives... In this case, the record is used

∫

f(x)dx

where is the sign ∫ is called the integral sign, the function f(x) Is the integrand, and f(x)dx - an integrand.

So if F(x) Is some kind of antiderivative for f(x) , then

∫

f(x)dx = F(x) +C

where C - an arbitrary constant (constant).

To understand the meaning of the set of antiderivatives of a function as an indefinite integral, the following analogy is appropriate. Let there be a door (traditional wooden door). Its function is to "be the door". What is the door made of? Made of wood. This means that the set of antiderivatives of the integrand "to be a door", that is, its indefinite integral, is the function "to be a tree + C", where C is a constant, which in this context can mean, for example, a tree species. Just like a door is made of wood with some tools, the derivative of a function is "made" from an antiderivative function using the formula that we learned by 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 the part of the problems of finding the indefinite integral, such integrands are given that, without special considerations, can be integrated directly, that is, according to the table of indefinite integrals. In more complicated problems, the integrand must first be transformed so that tabular 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 + С. 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 differentiation 4 or 3, or any other constant vanish.

Let us pose the integration problem: for this function f(x) find such a function F(x), whose derivative is 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 the antiderivative for the function f(x) if the derivative F(x) is equal to f(x), or, which is the same thing, the differential F(x) is equal to f(x) dx, i.e.

(2)

Therefore, a function is an antiderivative for a function. However, it is not the only antiderivative for. They also serve as functions

where WITH Is an arbitrary constant. This can be verified by differentiation.

Thus, if there is one antiderivative for a function, then for it there is an infinite number of 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) Is the antiderivative for the function f(x) on some interval NS, then any other antiderivative for f(x) on the same interval can be represented as F(x) + C, where WITH Is an arbitrary constant.

In the next example, we are already referring to the table of integrals, which will be given in Section 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 the integration in their entirety.

Example 2. Find sets of antiderivatives:

Solution. We find a set 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, and we will study the entire table of indefinite integrals 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 by formula (7) at n= -1/4 find

The integral is not the function itself f, and its product by the differential dx... This is done primarily to indicate which variable is being searched for the antiderivative. For example,

, ;

here in both cases the integrand is equal, but its indefinite integrals in the considered cases 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 the integration of this function.

The geometric meaning of the indefinite integral

Let it be required to find a curve y = F (x) and we already know that the tangent of the angle of inclination of the tangent at each of its points is a given function f (x) abscissa of this point.

According to the geometric meaning of the derivative, the tangent of the angle of inclination of the tangent at a given point of the curve y = F (x) is equal to the value of the derivative F "(x)... Hence, we need to find such a function F (x), for which F "(x) = f (x)... Function required in the task F (x) is the antiderivative of f (x)... The condition of the problem is satisfied not by one curve, but by a family of curves. y = F (x) is one of these curves, and any other curve can be obtained from it by parallel translation 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 is determined by an arbitrary constant (constant) of integration C.

Indefinite integral properties

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) is equal to the function f(x) up to a constant term , i.e.

(3)

Theorems 1 and 2 show that differentiation and integration are reciprocal operations.

Fact 6. Theorem 3. The constant factor in the integrand can be taken out of the indefinite integral sign , i.e.

The main task of differential calculus is finding the derivative f '(x) or differential df =f '(x)dx function f (x). In integral calculus, the inverse problem is solved. By a given function f (x) it is required to find such a function F (x), what F '(x) =f (x) or dF (x) =F '(x)dx =f (x)dx.

Thus, the main task of integral calculus is the restoration of function F (x) with respect to the known derivative (differential) of this function. Integral calculus has numerous applications in geometry, mechanics, physics, and engineering. It provides a general method for finding areas, volumes, centers of gravity, etc.

Definition. FunctionF (x), is called the antiderivative for the functionf (x) on the set X if it is differentiable for any andF '(x) =f (x) ordF (x) =f (x)dx.

Theorem. Any continuous on the segment [a;b] functionf (x) has the antiderivative on this segmentF (x).

Theorem. IfF 1 (x) andF 2 (x) - two different antiderivatives of the same functionf (x) on the set x, then they differ from each other by a constant term, i.e.F 2 (x) =F 1x) +C, where C is a constant.

Indefinite integral, its properties.

Definition. The aggregateF (x) +C of all antiderivatives of the functionf (x) on the set X is called an indefinite integral and is denoted by:

In formula (1) f (x)dx called integrand,f (x) is the integrand, x is the variable of integration, a С - constant of integration.

Consider the properties of the indefinite integral following from its definition.

1. The derivative of the indefinite integral is equal to the integrand, the differential of the indefinite integral is equal to the integrand:

2. The indefinite integral of the differential of some function is equal to the sum of this function and an arbitrary constant:

3. The constant factor a (a ≠ 0) can be taken outside the indefinite integral sign:

4. The indefinite integral of the algebraic sum of a finite number of functions is equal to the algebraic sum of integrals of these functions:

5. IfF (x) is the antiderivative of the functionf (x), then:

6 (invariance of the integration formulas). Any integration formula retains its form if the integration variable is replaced by any differentiable function of this variable:

whereu is a differentiable function.

Indefinite integrals table.

Let us give basic rules for integrating functions.

Let us give table of basic indefinite integrals.(Note that here, as in differential calculus, the letter u can denote as an independent variable (u =x) and a function of the independent variable (u =u (x)).)

Integrals 1 - 17 are called tabular.

Some of the above formulas of the table of integrals that have no analogue in the table of derivatives are checked by differentiating their right-hand sides.

Variable change and integration by parts in the indefinite integral.

Integration by substitution (variable replacement). Let it be required to calculate the integral

Theorem. Let the functionx = φ (t) is defined and differentiable on some set T and let X be the set of values ​​of this function, on which the functionf (x). Then if on the set X the functionf (