3 antiderivative of function. Antiderivative and integrals

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 derived from 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.

V following example we already turn 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 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, 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.

One of the operations of differentiation is finding the derivative (differential) and applying it to the study of functions.

The inverse problem is no less important. If the behavior of a function in the vicinity of each point of its definition is known, then how to restore the function as a whole, i.e. in the entire area of ​​its definition. This problem is the subject of study of the so-called integral calculus.

Integration is the opposite of differentiation. Or recovery of the function f (x) from a given derivative f` (x). The Latin word “integro” means restoration.

Example # 1.

Let (f (x)) '= 3x 2. Find f (x).

Solution:

Based on the rule of differentiation, it is easy to guess that f (x) = x 3, because

(x 3) '= 3x 2 However, you can easily notice that f (x) is found ambiguously. As f (x), you can take f (x) = x 3 +1 f (x) = x 3 +2 f (x) = x 3 -3, etc.

Because the derivative of each of them is equal to 3x 2. (The derivative of the constant is 0). All these functions differ from each other in a constant term. That's why common decision problems can be written in the form f (x) = x 3 + C, where C is any constant real number.

Any of the found functions f (x) is called antiderivative for the function F` (x) = 3x 2

Definition.

The function F (x) is called the antiderivative for the function f (x) on a given interval J if for all x from this interval F` (x) = f (x). So the function F (x) = x 3 is the antiderivative for f (x) = 3x 2 on (- ∞; ∞). Since, for all x ~ R, the equality is true: F` (x) = (x 3) `= 3x 2

As we have already noted, this function has an infinite number of antiderivatives.

Example # 2.

The function is the antiderivative for all on the interval (0; + ∞), since for all h from this interval, equality holds.

The integration problem is to find all its antiderivatives for a given function. In solving this problem, the following statement plays an important role:

A sign of the constancy of the function. If F "(x) = 0 on some interval I, then the function F is constant on this interval.

Proof.

We fix some x 0 from the interval I. Then, for any number x from such an interval, by virtue of the Lagrange formula, one can indicate a number c between x and x 0 such that

F (x) - F (x 0) = F "(c) (x-x 0).

By hypothesis, F ’(c) = 0, since c ∈1, therefore,

F (x) - F (x 0) = 0.

So, for all x from the interval I

that is, the function F remains constant.

All the antiderivatives of the function f can be written using one formula, which is called general form of antiderivatives for a function f. The following theorem is true ( the main property of antiderivatives):

Theorem. Any antiderivative for the function f on the interval I can be written as

F (x) + C, (1) where F (x) is one of the antiderivatives for the function f (x) on the interval I, and C is an arbitrary constant.

Let us explain this statement, in which two properties of the antiderivative are briefly formulated:

1. whatever number we put in expression (1) instead of С, we get the antiderivative for f on the interval I;
2. no matter what antiderivative Φ for f we take on the interval I, we can choose a number C such that for all x from the interval I the equality

Proof.

1. By hypothesis, the function F is an antiderivative for f on the interval I. Therefore, F "(x) = f (x) for any x∈1, therefore (F (x) + C)" = F "(x) + C" = f (x) + 0 = f (x), that is, F (x) + C is the antiderivative for the function f.
2. Let Ф (х) be one of the antiderivatives for the function f on the same interval I, that is, Ф "(x) = f (х) for all x∈I.

Then (Ф (x) - F (x)) "= Ф" (x) -F '(x) = f (x) -f (x) = 0.

Hence it follows in. the strength of the sign of the constancy of the function that the difference Ф (х) - F (х) is a function that takes some constant value C on the interval I.

Thus, for all x from the interval I, the equality Φ (x) - F (x) = C is true, as required. The main property of the antiderivative can be given a geometric meaning: the graphs of any two antiderivatives for the function f are obtained from each other by parallel translation along the Oy axis

Questions for notes

The function F (x) is the antiderivative for the function f (x). Find F (1) if f (x) = 9x2 - 6x + 1 and F (-1) = 2.

Find all antiderivatives for a function

For function (x) = cos2 * sin2x, find the antiderivative F (x) if F (0) = 0.

For the function, find the antiderivative whose graph passes through the point

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 is to crochet something useful in the form of an integral icon from hard-to-reach places then welcome! Learn how to solve integrals and why you can't do without it.

We study the concept of "integral"

Integration has been known since 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 still need a basic knowledge of the basics of mathematical analysis. It is these fundamental information about you you will find on 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 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:

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 infinitely a large number infinitesimal 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.

Bari Alibasov and the "Integral" group

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Integral computation rules for dummies

Indefinite integral properties

How to solve an 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 a few examples of finding indefinite integrals... We suggest that you 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. Ask and they will tell you everything they know about calculating integrals. With our help, any triple or curvilinear integral on a closed surface will be within your power.

Antiderivative

Defining an Antiderivative Function

• Function y = F (x) called the antiderivative for the function y = f (x) at a given interval NS, if for all NS ∈NS equality holds: F ′ (x) = f (x)

It can be read in two ways:

1. f derivative of a function F
2. F antiderivative for function f

The property of antiderivatives

• If F (x)- antiderivative for function f (x) on a given interval, then the function f (x) has infinitely many antiderivatives, and all these antiderivatives can be written as F (x) + C, where C is an arbitrary constant.

Geometric interpretation

• Graphs of all antiderivatives of a given function f (x) are obtained from the graph of any one antiderivative by parallel translations along the O axis at.

Antiderivatives calculation rules

1. The antiderivative of the sum is equal to the sum of the antiderivatives... If F (x)- antiderivative for f (x), and G (x) is the antiderivative for g (x), then F (x) + G (x)- antiderivative for f (x) + g (x).
2. The constant factor can be moved outside the sign of the derivative... If F (x)- antiderivative for f (x), and k- constant, then k F (x)- antiderivative for k f (x).
3. If F (x)- antiderivative for f (x), and k, b- permanent, moreover k ≠ 0, then 1 / k F (kx + b)- antiderivative for f (kx + b).

Remember!

Any function F (x) = x 2 + C , where C is an arbitrary constant, and only such a function is the antiderivative for the function f (x) = 2x.

• For example:

  F "(x) = (x 2 + 1)" = 2x = f (x);

  f (x) = 2x, since F "(x) = (x 2 - 1)" = 2x = f (x);

  f (x) = 2x, since F "(x) = (х 2 –3)" = 2x = f (x);

The relationship between the graphs of a function and its antiderivative:

1. If the graph of the function f (x)> 0 F (x) increases in this interval.
2. If the graph of the function f (x)<0 on the interval, then the graph of its antiderivative F (x) decreases in this interval.
3. If f (x) = 0, then the graph of its antiderivative F (x) at this point changes from increasing to decreasing (or vice versa).

To denote the antiderivative, the sign of an indefinite integral is used, that is, an integral without indicating the limits of integration.

Indefinite integral

Definition:

• An indefinite integral of a function f (x) is the expression F (x) + C, that is, the collection of all antiderivatives of a given function f (x). The indefinite integral is denoted as follows: \ int f (x) dx = F (x) + C

• f (x)- called the integrand;
• f (x) dx- called the integrand;
• x- called the variable of integration;
• F (x)- one of the antiderivatives of the function f (x);
• WITH is an arbitrary constant.

Indefinite integral properties

1. The derivative of the indefinite integral is equal to the integrand: (\ int f (x) dx) \ prime = f (x).
2. The constant factor of the integrand can be taken outside the integral sign: \ int k \ cdot f (x) dx = k \ cdot \ int f (x) dx.
3. The integral of the sum (difference) of functions is equal to the sum (difference) of the integrals of these functions: \ int (f (x) \ pm g (x)) dx = \ int f (x) dx \ pm \ int g (x) dx.
4. If k, b are constants, and k ≠ 0, then \ int f (kx + b) dx = \ frac (1) (k) \ cdot F (kx + b) + C.

Table of antiderivatives and indefinite integrals

Function f (x)	Antiderivative F (x) + C	Indefinite integrals \ int f (x) dx = F (x) + C
0	C	\ int 0 dx = C
f (x) = k	F (x) = kx + C	\ int kdx = kx + C
f (x) = x ^ m, m \ not = -1	F (x) = \ frac (x ^ (m + 1)) (m + 1) + C	\ int x (^ m) dx = \ frac (x ^ (m + 1)) (m + 1) + C
f (x) = \ frac (1) (x)	F (x) = l n \ lvert x \ rvert + C	\ int \ frac (dx) (x) = l n \ lvert x \ rvert + C
f (x) = e ^ x	F (x) = e ^ x + C	\ int e (^ x) dx = e ^ x + C
f (x) = a ^ x	F (x) = \ frac (a ^ x) (l na) + C	\ int a (^ x) dx = \ frac (a ^ x) (l na) + C
f (x) = \ sin x	F (x) = - \ cos x + C	\ int \ sin x dx = - \ cos x + C
f (x) = \ cos x	F (x) = \ sin x + C	\ int \ cos x dx = \ sin x + C
f (x) = \ frac (1) (\ sin (^ 2) x)	F (x) = - \ ctg x + C	\ int \ frac (dx) (\ sin (^ 2) x) = - \ ctg x + C
f (x) = \ frac (1) (\ cos (^ 2) x)	F (x) = \ tg x + C	\ int \ frac (dx) (\ sin (^ 2) x) = \ tg x + C
f (x) = \ sqrt (x)	F (x) = \ frac (2x \ sqrt (x)) (3) + C
f (x) = \ frac (1) (\ sqrt (x))	F (x) = 2 \ sqrt (x) + C
f (x) = \ frac (1) (\ sqrt (1-x ^ 2))	F (x) = \ arcsin x + C	\ int \ frac (dx) (\ sqrt (1-x ^ 2)) = \ arcsin x + C
f (x) = \ frac (1) (\ sqrt (1 + x ^ 2))	F (x) = \ arctan x + C	\ int \ frac (dx) (\ sqrt (1 + x ^ 2)) = \ arctg x + C
f (x) = \ frac (1) (\ sqrt (a ^ 2-x ^ 2))	F (x) = \ arcsin \ frac (x) (a) + C	\ int \ frac (dx) (\ sqrt (a ^ 2-x ^ 2)) = \ arcsin \ frac (x) (a) + C
f (x) = \ frac (1) (\ sqrt (a ^ 2 + x ^ 2))	F (x) = \ arctg \ frac (x) (a) + C	\ int \ frac (dx) (\ sqrt (a ^ 2 + x ^ 2)) = \ frac (1) (a) \ arctg \ frac (x) (a) + C
f (x) = \ frac (1) (1 + x ^ 2)	F (x) = \ arctg + C	\ int \ frac (dx) (1 + x ^ 2) = \ arctg + C
f (x) = \ frac (1) (\ sqrt (x ^ 2-a ^ 2)) (a \ not = 0)	F (x) = \ frac (1) (2a) l n \ lvert \ frac (x-a) (x + a) \ rvert + C	\ int \ frac (dx) (\ sqrt (x ^ 2-a ^ 2)) = \ frac (1) (2a) l n \ lvert \ frac (x-a) (x + a) \ rvert + C
f (x) = \ tg x	F (x) = - l n \ lvert \ cos x \ rvert + C	\ int \ tg x dx = - l n \ lvert \ cos x \ rvert + C
f (x) = \ ctg x	F (x) = l n \ lvert \ sin x \ rvert + C	\ int \ ctg x dx = l n \ lvert \ sin x \ rvert + C
f (x) = \ frac (1) (\ sin x)	F (x) = l n \ lvert \ tg \ frac (x) (2) \ rvert + C	\ int \ frac (dx) (\ sin x) = l n \ lvert \ tg \ frac (x) (2) \ rvert + C
f (x) = \ frac (1) (\ cos x)	F (x) = l n \ lvert \ tg (\ frac (x) (2) + \ frac (\ pi) (4)) \ rvert + C	\ int \ frac (dx) (\ cos x) = l n \ lvert \ tg (\ frac (x) (2) + \ frac (\ pi) (4)) \ rvert + C

Newton-Leibniz formula

Let be f (x) given function, F its arbitrary antiderivative.

\ int_ (a) ^ (b) f (x) dx = F (x) | _ (a) ^ (b)= F (b) - F (a)

where F (x)- antiderivative for f (x)

That is, the integral of the function f (x) on the interval is equal to the difference of antiderivatives at the points b and a.

Curved trapezoid area

Curved trapezoid is a figure bounded by the graph of a non-negative and continuous function on a segment f, the Ox axis and straight lines x = a and x = b.

The area of ​​a curved trapezoid is found by the Newton-Leibniz formula:

S = \ int_ (a) ^ (b) f (x) dx

Function F (x ) called antiderivative for function f (x) at a given interval, if for all x from this interval, the equality

F "(x ) = f(x ) .

For example, the function F (x) = x 2 f (x ) = 2NS , because

F "(x) = (x 2 )" = 2x = f (x). ◄

The main property of the antiderivative

If F (x) - antiderivative for function f (x) on a given interval, then the function f (x) has infinitely many antiderivatives, and all these antiderivatives can be written as F (x) + C, where WITH Is an arbitrary constant.

For example. Function F (x) = x 2 + 1 is the antiderivative of the function f (x ) = 2NS , because F "(x) = (x 2 + 1 )" = 2 x = f (x); function F (x) = x 2 - 1 is the antiderivative of the function f (x ) = 2NS , because F "(x) = (x 2 - 1)" = 2x = f (x) ; function F (x) = x 2 - 3 is the antiderivative of the function f (x) = 2NS , because F "(x) = (x 2 - 3)" = 2 x = f (x); any function F (x) = x 2 + WITH , where WITH - an arbitrary constant, and only such a function is an antiderivative for the function f (x) = 2NS . ◄

Antiderivatives calculation rules

1. If F (x) - antiderivative for f (x) , a G (x) - antiderivative for g (x) , then F (x) + G (x) - antiderivative for f (x) + g (x) ... In other words, the antiderivative of the sum is equal to the sum of the antiderivatives .
2. If F (x) - antiderivative for f (x) , and k - constant, then k · F (x) - antiderivative for k · f (x) ... In other words, the constant factor can be moved outside the sign of the derivative .
3. If F (x) - antiderivative for f (x) , and k,b- permanent, moreover k ≠ 0 , then 1 / k F ( k x + b ) - antiderivative for f(k x + b) .

Indefinite integral

Indefinite integral from function f (x) called expression F (x) + C, that is, the totality of all antiderivatives of a given function f (x) ... The indefinite integral is denoted as follows:

∫ f (x) dx = F (x) + С ,

f (x)- call the integrand ;

f (x) dx- called integrand ;

x - called variable of integration ;

F (x) - one of the antiderivatives of the function f (x) ;

WITH Is an arbitrary constant.

For example, ∫ 2 x dx =NS 2 + WITH , ∫ cosx dx = sin NS + WITH etc. ◄

The word "integral" comes from the Latin word integer which means "restored". Considering the indefinite integral of 2 x, we sort of restore the function NS 2 whose derivative is equal to 2 x... Reconstruction of a function from its derivative, or, which is the same, finding an indefinite integral over a given integrand, is called integrating this function. Integration is the inverse of differentiation. In order to check whether the integration is correct, it is enough to differentiate the result and obtain the integrand function.

Basic properties of the indefinite integral

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

(∫ f (x) dx )" = f (x) .

2. The constant factor of the integrand can be taken outside the integral sign:

∫ k · f (x) dx = k · ∫ f (x) dx .

3. The integral of the sum (difference) of functions is equal to the sum (difference) of the integrals of these functions:

∫ ( f (x) ± g (x ) ) dx = ∫ f (x) dx ± ∫ g (x ) dx .

4. If k,b- permanent, moreover k ≠ 0 , then

∫ f ( k x + b) dx = 1 / k F ( k x + b ) + C .

Table of antiderivatives and indefinite integrals

	f (x)	F (x) + C	∫ f (x) dx = F (x) + С
I.	$$0$$	$$ C $$	$$ \ int 0dx = C $$
II.	$$ k $$	$$ kx + C $$	$$ \ int kdx = kx + C $$
III.	$$ x ^ n ~ (n \ neq-1) $$	$$ \ frac (x ^ (n + 1)) (n + 1) + C $$	$$ \ int x ^ ndx = \ frac (x ^ (n + 1)) (n + 1) + C $$
IV.	$$ \ frac (1) (x) $$	$$ \ ln | x | + C $$	$$ \ int \ frac (dx) (x) = \ ln | x | + C $$
V.	$$ \ sin x $$	$$ - \ cos x + C $$	$$ \ int \ sin x ~ dx = - \ cos x + C $$
Vi.	$$ \ cos x $$	$$ \ sin x + C $$	$$ \ int \ cos x ~ dx = \ sin x + C $$
Vii.	$$ \ frac (1) (\ cos ^ 2x) $$	$$ \ textrm (tg) ~ x + C $$	$$ \ int \ frac (dx) (\ cos ^ 2x) = \ textrm (tg) ~ x + C $$
VIII.	$$ \ frac (1) (\ sin ^ 2x) $$	$$ - \ textrm (ctg) ~ x + C $$	$$ \ int \ frac (dx) (\ sin ^ 2x) = - \ textrm (ctg) ~ x + C $$
IX.	$$ e ^ x $$	$$ e ^ x + C $$	$$ \ int e ^ xdx = e ^ x + C $$
X.	$$ a ^ x $$	$$ \ frac (a ^ x) (\ ln a) + C $$	$$ \ int a ^ xdx = \ frac (a ^ x) (\ ln a) + C $$
XI.	$$ \ frac (1) (\ sqrt (1-x ^ 2)) $$	$$ \ arcsin x + C $$	$$ \ int \ frac (dx) (\ sqrt (1-x ^ 2)) = \ arcsin x + C $$
XII.	$$ \ frac (1) (\ sqrt (a ^ 2-x ^ 2)) $$	$$ \ arcsin \ frac (x) (a) + C $$	$$ \ int \ frac (dx) (\ sqrt (a ^ 2-x ^ 2)) = \ arcsin \ frac (x) (a) + C $$
XIII.	$$ \ frac (1) (1 + x ^ 2) $$	$$ \ textrm (arctg) ~ x + C $$	$$ \ int \ frac (dx) (1 + x ^ 2) = \ textrm (arctg) ~ x + C $$
XIV.	$$ \ frac (1) (a ^ 2 + x ^ 2) $$	$$ \ frac (1) (a) \ textrm (arctg) ~ \ frac (x) (a) + C $$	$$ \ int \ frac (dx) (a ^ 2 + x ^ 2) = \ frac (1) (a) \ textrm (arctg) ~ \ frac (x) (a) + C $$
XV.	$$ \ frac (1) (\ sqrt (a ^ 2 + x ^ 2)) $$	$$ \ ln | x + \ sqrt (a ^ 2 + x ^ 2) | + C $$	$$ \ int \ frac (dx) (\ sqrt (a ^ 2 + x ^ 2)) = \ ln | x + \ sqrt (a ^ 2 + x ^ 2) | + C $$
XVI.	$$ \ frac (1) (x ^ 2-a ^ 2) ~ (a \ neq0) $$	$$ \ frac (1) (2a) \ ln \ begin (vmatrix) \ frac (x-a) (x + a) \ end (vmatrix) + C $$	$$ \ int \ frac (dx) (x ^ 2-a ^ 2) = \ frac (1) (2a) \ ln \ begin (vmatrix) \ frac (xa) (x + a) \ end (vmatrix) + C $$
XVII.	$$ \ textrm (tg) ~ x $$	$$ - \ ln | \ cos x | + C $$	$$ \ int \ textrm (tg) ~ x ~ dx = - \ ln | \ cos x | + C $$
XVIII.	$$ \ textrm (ctg) ~ x $$	$$ \ ln | \ sin x | + C $$	$$ \ int \ textrm (ctg) ~ x ~ dx = \ ln | \ sin x | + C $$
XIX.	$$ \ frac (1) (\ sin x) $$	$$ \ ln \ begin (vmatrix) \ textrm (tg) ~ \ frac (x) (2) \ end (vmatrix) + C $$	$$ \ int \ frac (dx) (\ sin x) = \ ln \ begin (vmatrix) \ textrm (tg) ~ \ frac (x) (2) \ end (vmatrix) + C $$
XX.	$$ \ frac (1) (\ cos x) $$	$$ \ ln \ begin (vmatrix) \ textrm (tg) \ left (\ frac (x) (2) + \ frac (\ pi) (4) \ right) \ end (vmatrix) + C $$	$$ \ int \ frac (dx) (\ cos x) = \ ln \ begin (vmatrix) \ textrm (tg) \ left (\ frac (x) (2) + \ frac (\ pi) (4) \ right ) \ end (vmatrix) + C $$
Antiderivatives and indefinite integrals given in this table are usually called tabular antiderivatives and tabular integrals .

Definite integral

Let in the interval [a; b] continuous function is given y = f (x) , then definite integral from a to b function f (x) is called the increment of the antiderivative F (x) this function, that is

$$ \ int_ (a) ^ (b) f (x) dx = F (x) | (_a ^ b) = ~~ F (a) -F (b). $$

The numbers a and b are named accordingly lower and top limits of integration.

Basic rules for calculating a definite integral

1. \ (\ int_ (a) ^ (a) f (x) dx = 0 \);

2. \ (\ int_ (a) ^ (b) f (x) dx = - \ int_ (b) ^ (a) f (x) dx \);

3. \ (\ int_ (a) ^ (b) kf (x) dx = k \ int_ (a) ^ (b) f (x) dx, \) where k - constant;

4. \ (\ int_ (a) ^ (b) (f (x) ± g (x)) dx = \ int_ (a) ^ (b) f (x) dx ± \ int_ (a) ^ (b) g (x) dx \);

5. \ (\ int_ (a) ^ (b) f (x) dx = \ int_ (a) ^ (c) f (x) dx + \ int_ (c) ^ (b) f (x) dx \);

6. \ (\ int _ (- a) ^ (a) f (x) dx = 2 \ int_ (0) ^ (a) f (x) dx \), where f (x) - even function;

7. \ (\ int _ (- a) ^ (a) f (x) dx = 0 \), where f (x) Is an odd function.

Comment ... In all cases, it is assumed that the integrands are integrable on numerical intervals, the boundaries of which are the limits of integration.

Geometric and physical meaning of a definite integral

Geometric meaning definite integral	Physical sense definite integral
Square S curvilinear trapezoid (a figure bounded by the graph of a continuous positive in the interval [a; b] function f (x) , axis Ox and direct x = a , x = b ) is calculated by the formula $$ S = \ int_ (a) ^ (b) f (x) dx. $$	Way s, which the material point has overcome, moving in a straight line with a speed varying according to the law v (t) , for the time interval a ; b], then the area of ​​the figure, limited by the graphs of these functions and straight lines x = a , x = b , calculated by the formula $$ S = \ int_ (a) ^ (b) (f (x) -g (x)) dx. $$
	For example. Calculate the area of ​​the figure bounded by the lines y = x 2 and y = 2- x . Let us depict schematically the graphs of these functions and highlight in a different color the figure whose area is to be found. To find the limits of integration, we solve the equation: x 2 = 2- x ; x 2 + x - 2 = 0 ; x 1 = -2, x 2 = 1 . $$ S = \ int _ (- 2) ^ (1) ((2-x) -x ^ 2) dx = $$
$$ = \ int _ (- 2) ^ (1) (2-xx ^ 2) dx = \ left (2x- \ frac (x ^ 2) (2) - \ frac (x ^ 3) (2) \ right ) \ bigm | (_ (- 2) ^ (~ 1)) = 4 \ frac (1) (2). $$ ◄

Volume of a body of revolution

	If the body is obtained as a result of rotation about the axis Ox curvilinear trapezoid, limited by the graph of continuous and non-negative in the interval [a; b] function y = f (x) and direct x = a and x = b then it is called body of revolution . The volume of a body of revolution is calculated by the formula $$ V = \ pi \ int_ (a) ^ (b) f ^ 2 (x) dx. $$ If the body of revolution is obtained as a result of the rotation of a figure bounded above and below by the graphs of functions y = f (x) and y = g (x) , respectively, then $$ V = \ pi \ int_ (a) ^ (b) (f ^ 2 (x) -g ^ 2 (x)) dx. $$
	For example. We calculate the volume of a cone with a radius r and height h . Place the cone in a rectangular coordinate system so that its axis coincides with the axis Ox , and the center of the base was at the origin. Generator rotation AB defines a cone. Since the equation AB $$ \ frac (x) (h) + \ frac (y) (r) = 1, $$ $$ y = r- \ frac (rx) (h) $$
and for the volume of the cone we have $$ V = \ pi \ int_ (0) ^ (h) (r- \ frac (rx) (h)) ^ 2dx = \ pi r ^ 2 \ int_ (0) ^ (h) (1- \ frac ( x) (h)) ^ 2dx = - \ pi r ^ 2h \ cdot \ frac ((1- \ frac (x) (h)) ^ 3) (3) | (_0 ^ h) = - \ pi r ^ 2h \ left (0- \ frac (1) (3) \ right) = \ frac (\ pi r ^ 2h) (3). $$ ◄