Antiderivative function and indefinite integral

Fact 1. Integration is the opposite of 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 set 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 slope 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.

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.

In this article, we will list the main properties of the definite integral. Most of these properties are proved based on the concepts of the definite integral of Riemann and Darboux.

The definition of a definite integral is very often done using the first five properties, so we will refer to them when necessary. The rest of the properties of a definite integral are mainly used to evaluate various expressions.

Before moving on to the basic properties of the definite integral, let us agree that a does not exceed b.

For the function y = f (x), defined at x = a, equality is true.

That is, the value of a definite integral with coinciding limits of integration is zero. This property is a consequence of the definition of the Riemann integral, since in this case each integral sum for any partition of the interval and any choice of points is equal to zero, since, therefore, the limit of the integral sums is zero.

For a function integrable on a segment, .

In other words, when changing the upper and lower limits of integration in places, the value of the definite integral changes to the opposite. This property of a definite integral also follows from the concept of the Riemann integral, only the numbering of the partition of a segment should be started from the point x = b.

for functions y = f (x) and y = g (x) integrable on an interval.

Proof.

We write the integral sum of the function for a given division of a segment and a given choice of points:

where and are the integral sums of the functions y = f (x) and y = g (x) for the given partition of the segment, respectively.

Passing to the limit at we obtain that, by the definition of the Riemann integral, it is equivalent to the assertion of the property being proved.

A constant factor can be taken out of the sign of a definite integral. That is, for a function y = f (x) integrable on an interval and an arbitrary number k, the equality .

The proof of this property of a definite integral is absolutely similar to the previous one:


Let the function y = f (x) be integrable on the interval X, and and then .

This property is true for both, and for or.

The proof can be carried out using the previous properties of the definite integral.

If a function is integrable on a segment, then it is integrable on any inner segment as well.

The proof is based on the property of Darboux sums: if you add new points to the existing partition of the segment, then the lower Darboux sum will not decrease, and the upper one will not increase.

If the function y = f (x) is integrable on an interval and for any value of the argument, then .

This property is proved through the definition of the Riemann integral: any integral sum for any choice of partition points of a segment and points at will be non-negative (not positive).

Consequence.

For functions y = f (x) and y = g (x) integrable on an interval, the following inequalities hold:


This statement means that the integration of inequalities is admissible. We will use this corollary to prove the following properties.

Let the function y = f (x) be integrable on an interval, then the inequality .

Proof.

It's obvious that ... In the previous property, we found out that the inequality can be integrated term by term, therefore, it is true ... This double inequality can be written as .

Let the functions y = f (x) and y = g (x) be integrable on an interval and for any value of the argument, then , where and .

The proof is similar. Since m and M are the smallest and largest values ​​of the function y = f (x) on the segment, then ... Multiplying the double inequality by the nonnegative function y = g (x) leads us to the following double inequality. Integrating it on a segment, we arrive at the assertion being proved.

Consequence.

If we take g (x) = 1, then the inequality takes the form .

First average value formula.

Let the function y = f (x) be integrable on an interval, and then there is a number such that .

Consequence.

If the function y = f (x) is continuous on an interval, then there is a number such that .

The first formula for the mean in generalized form.

Let the functions y = f (x) and y = g (x) be integrable on an interval, and, and g (x)> 0 for any value of the argument. Then there is a number such that .

Second formula for the average.

If the function y = f (x) is integrable on an interval and y = g (x) is monotone, then there exists a number such that the equality .

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 of an integral you know of is to crochet something useful from hard-to-reach places with a crochet in the shape of an integral icon, then you are welcome! Learn how to solve elementary and other integrals and why you can't do without it in mathematics.

Exploring the concept « integral »

Integration has been known since 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 still need a basic knowledge of the basics of calculus. We already have information about limits and derivatives necessary for understanding integrals in our blog.

Indefinite integral

Suppose we have some kind of function f (x) .

Indefinite integral of a function f (x) such a function is called F (x) whose derivative is equal to the function f (x) .

In other words, the integral is the reverse derivative or antiderivative. By the way, read about how to calculate derivatives in our article.

The antiderivative exists for all continuous functions. Also, the sign of a constant 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 the antiderivatives of elementary functions, it is convenient to bring them down to 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 ​​a figure, the mass of an inhomogeneous body, the path traveled with uneven movement, and much more. It should be remembered that the integral is the sum of an infinitely large number of infinitely small terms.

As an example, let's imagine a graph of some function.

How to find the area of ​​a shape bounded by the graph of a function? Using the integral! We divide the curvilinear trapezoid, bounded by the coordinate axes and the graph of the function, into infinitely small segments. Thus, 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 the limits of integration.

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Integral computation rules for dummies

Indefinite integral properties

How to solve indefinite integral? Here we will look at the properties of the indefinite integral, which will come in handy 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. It is also true for the difference:

Properties of the definite integral

• Linearity:

• The integral sign changes if the integration limits are reversed:

• At any points a, b and with:

We have already found out that the definite integral is the limit of the sum. But how do you get a specific value when solving an example? For this, there is the Newton-Leibniz formula:

Integral solutions examples

Below we will consider an indefinite integral and examples with a solution. We offer you to independently figure out the intricacies of the solution, and if something is not clear, ask questions in the comments.

To consolidate the material, watch the video on how integrals are solved in practice. Don't be discouraged if the integral isn't given right away. Contact the professional student service and you can handle any triple or curvilinear integral over a closed surface.

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 functions 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]. 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.

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

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

2. The 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 out of 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

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 equations 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 using 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)