When solving geometric tasks in space, the problem of determining the distance between the plane and the point occurs. In some cases it is necessary for a comprehensive solution. This magnitude can be calculated if you find a projection on the point plane. Consider this question more in the article.

Equation to describe the plane

Before moving to the consideration of the question regarding how to find the projection of the point to the plane, you should get acquainted with the types of equations that define the last in three-dimensional space. Read more - below.

The equation of a shared form determining all the points that belong to this plane is the following:

A * X + B * Y + C * Z + D \u003d 0.

The first three coefficients are the coordinates of the vector, which is called the guide for the plane. It coincides with the normal for her, that is, is perpendicular. This vector is n¯ (a; b; c). The free coefficient D is uniquely determined from the knowledge of the coordinates of any point belonging to the plane.

The concept of the projection of the point and its calculation

Suppose that a certain point p (x 1; y 1; z 1) and a plane are specified. It is determined by the equation in general form. If you have a perpendicular straight line from P to a given plane, it is obvious that it will cross the latter in one specific point Q (x 2; Y 2; Z 2). Q is called the projection P on the plane under consideration. The length of the PQ segment is called the distance from the point P to the plane. Thus, PQ itself is a perpendicular plane.

How can I find the coordinates of the projection point to the plane? Make it is not difficult. To begin with, the equation is direct, which will be perpendicular to the plane. It will belong to POP. Since the normal vector n¯ (a; b; c) of this direct must be parallel, the equation for it in the appropriate form will be recorded as follows:

(x; y; z) \u003d (x 1; y 1; z 1) + λ * (a; b; c).

Where λ - valid numberwhich is customary called the parameter of the equation. By changing it, you can get any point direct.

After the vector equation is recorded for the perpendicular plane of the line, it is necessary to find a common intersection point for the geometric objects under consideration. Its coordinates will be a projection P. Because they must meet both equalities (for direct and for a plane), the task is reduced to solving the corresponding system of linear equations.

The concept of projection is often used when studying the drawings. They are depicted side and horizontal projections of the parts on the ZY, ZX, and XY planes.

Calculation of the distance from the plane to the point

As noted above, knowledge of the projection coordinates on the point plane allows to determine the distance between them. Using the designations introduced in the previous paragraph, we obtain that the desired distance is equal to the length of the PQ segment. To calculate it, it is sufficient to find the coordinates of the vector PQ¯, and then calculate its module according to the well-known formula. The final expression for D distances between P point and plane takes the form:

d \u003d | PQ¯ | \u003d √ ((x 2 - x 1) 2 + (y 2 - y 1) 2 + (z 2 - z 1) 2).

The resulting value of D is represented in units in which the current XYZ coordinate system is specified.

Example of the task

Suppose there is a point n (0; -2; 3) and the plane, which is described by the following equation:

You should find a projection point to the plane and calculate the distance between them.

First of all, the equation is direct, which crosses the plane at an angle of 90 o. We have:

(x; y; z) \u003d (0; -2; 3) + λ * (2; -1; 1).

By writing this equality explicit form, we arrive at the following system of equations:

Substituting the coordinate values \u200b\u200bof the first three equalities in the fourth, we obtain the value of λ, which determines the coordinates of the common point of the direct and plane:

2 * (2 * λ) - (-2 - λ) + λ + 3 + 4 \u003d 0 \u003d\u003e

6 * λ + 9 \u003d 0 \u003d\u003e

λ \u003d 9/6 \u003d 3/2 \u003d 1.5.

We substitute the found parameter in and find the coordinates of the projection of the initial point to the plane:

(x; y; z) \u003d (0; -2; 3) + 1.5 * (2; -1; 1) \u003d (3; -3,5; 4,5).

To calculate the distance between the task defined in the condition, geometric objects apply the formula for D:

d \u003d √ ((3 - 0) 2 + (-3,5 + 2) 2 + (4.5 - 3) 2) \u003d 3,674.

In this task, we showed how to find a projection of a point on an arbitrary plane and how to calculate the distance between them.

Integration is one of the main operations in the matanalize. The tables of known primary can be useful, but now they, after the appearance of computer algebra systems, lose their significance. Below is the list Most of all encountered.

Table of main integrals

Another compact option

Table of integrals from trigonometric functions

From rational functions

From irrational functions

Integrals from transcendental functions

"C" - an arbitrary integration constant, which is determined if the integral value is known at any point. Each function has an infinite number of primitive.

Most schoolchildren and students have problems with the calculation of integrals. On this page collected tables integrals From trigonometric, rational, irrational and transcendental functions that will help in solving. You still help the derivatives table.

Video - How to find integrals

If this topic is not quite clear, look at the video in which everything explains in detail.

\u003e\u003e Integration methods

Basic integration methods

Definition of the integral, a specific and indefinite integral, integral table, Newton-laboratory formula, integration in parts, examples of calculating integrals.

Uncertain integral

Function f (x), differentiable in this gap, is called perfect for function F (x), or by the integral from f (x), if for any x ∈x, equality is true:

F "(x) \u003d f (x). (8.1)

Finding all the primary for this feature is called it integration. Uncertain integral functionf (x) at this gap is called the set of all the primitive functions for the function f (x); Designation -

If f (x) is some kind of functional function f (x), then ∫ f (x) dx \u003d f (x) + c, (8.2)

where there is an arbitrary constant.

Table integrals

Directly from the definition we get basic properties not certain integral and a list of tabular integrals:

1) d∫f (x) dx \u003d f (x)

2) ∫df (x) \u003d f (x) + c

3) ∫af (x) dx \u003d a∫f (x) dx (a \u003d const)

4) ∫ (f (x) + g (x)) dx \u003d ∫f (x) dx + ∫g (x) dx

List of tabular integrals

1. ∫x m dx \u003d x m + 1 / (m + 1) + c; (m ≠ -1)

3.∫A X DX \u003d A X / LN A + C (A\u003e 0, A ≠ 1)

4.∫E x DX \u003d E X + C

5.∫sin x DX \u003d COSX + C

6.∫cos x DX \u003d - SIN X + C

7. \u003d ArctG X + C

8. \u003d Arcsin X + C

10. \u003d - CTG X + C

Replacing the variable

For the integration of many functions, the method of replacing a variable or substitutionsallowing to bring integrals to tabular form.

If the function f (z) is continuous to [α, β], the function z \u003d g (x) has a continuous derivative and α ≤ g (x) ≤ β, then

∫ f (g (x)) g "(x) dx \u003d ∫f (z) dz, (8.3)

moreover, after integration, the substitution Z \u003d G (X) should be made in the right part.

To prove, it is enough to write the source integral in the form:

∫ F (G (x)) g "(x) dx \u003d ∫ f (g (x)) DG (x).

For example:

1)

2) .

Integration method in parts

Let u \u003d f (x) and v \u003d g (x) be functions that are continuous. Then, by work,

d (UV)) \u003d UDV + VDU or UDV \u003d D (UV) - VDU.

For the expression D (UV), the first, obviously, will be UV, so the formula is:

∫ UDV \u003d UV - ∫ VDU (8.4.)

This formula expresses the rule integration in parts. It results in the integration of the expression UDV \u003d UV "DX to integrating the expression VDU \u003d VU" DX.

Let, for example, you need to find ∫xcosx dx. Put u \u003d x, dv \u003d cosxdx, so that du \u003d dx, v \u003d sinx. Then

∫xcosxdx \u003d ∫x d (sin x) \u003d x sin x - ∫sin x dx \u003d x sin x + cosx + c.

The integration rule in parts has a more limited scope than the replacement of the variable. But there are whole classes of integrals, for example,

∫x k ln m xDX, ∫x k sinbxdx, ∫ x k cosbxdx, ∫x k e ax and others that are calculated using integration in parts.

Certain integral

The concept of a specific integral is enhanced as follows. Let the F (X) function define on the segment. We break the segment [a, b] on n. parts dots a \u003d x 0< x 1 <...< x n = b. Из каждого интервала (x i-1 , x i) возьмем произвольную точку ξ i и составим сумму f(ξ i) Δx i где
Δ x i \u003d x i - x i-1. The sum of the form f (ξ i) Δ x I is called integral sum, and its limit at λ \u003d maxΔx i → 0, if it exists and is finite, called Certain integralfunctions f (x) from a. before b. And indicated:

F (ξ i) Δx i (8.5).

Function F (x) in this case is called integrable on cut, numbers a and b are called lower and upper integral limit.

For a specific integral, the following properties are valid:

4), (k \u003d const, k∈R);

5)

6)

7) F (ξ) (B - a) (ξ∈).

The last property is called Theorest on the average meaning.

Let F (x) be continuous on. Then there is an indefinite integral on this segment

∫f (x) dx \u003d f (x) + c

and takes place formula Newton Labitsa, binding a specific integral with uncertain:

F (B) - F (a). (8.6)

Geometric interpretation: A certain integral is an area of \u200b\u200bcurvilinear trapezium, limited from above the curve y \u003d f (x), straight x \u003d a and x \u003d b and the segment of the axis OX..

Invalid integrals

Integrals with infinite limits and integrals from discontinuous (unlimited) functions are called incompatible. Incompatible integrals of I kind - These are integrals at an infinite gap defined as follows:

(8.7)

If this limit exists and is finite, then called converging incomplete integral from F (X) on the interval [A, + ∞), and the function f (x) is called integrated at an infinite interval[A, + ∞). Otherwise about the integral say he does not exist or diverge.

In the same way, incomprehensible integrals at the intervals (-∞, b] and (-∞, + ∞) are determined:

We define the concept of integral from unlimited function. If f (x) is continuous for all values x. Cut, except for the point C, in which f (x) has an endless gap, then incompatible integral II genus from f (x) in the range from A to B The amount is called:

if these limits exist and are finite. Designation:

Examples of calculation of integrals

Example 3.30. Calculate ∫dx / (x + 2).

Decision. Denote by t \u003d x + 2, then dx \u003d dt, ∫dx / (x + 2) \u003d ∫dt / t \u003d ln | t | + C \u003d ln | x + 2 | + C.

Example 3.31.. Find ∫ TGXDX.

Decision.∫ TGXDX \u003d ∫SINX / COSXDX \u003d - ∫DCOSX / COSX. Let T \u003d Cosx, then ∫ TGXDX \u003d -∫ DT / T \u003d - LN | T | + C \u003d -LN | COSX | + C.

Example3.32 . Find ∫dx / sinx

Decision.

Example3.33. To find .

Decision. =

.

Example3.34 . Find ∫arctgxdx.

Decision. We integrate in parts. Denote u \u003d arctgx, dv \u003d dx. Then du \u003d dx / (x 2 +1), v \u003d x, from where ∫arctgxdx \u003d XARCTGX - ∫ XDX / (X 2 +1) \u003d XARCTGX + 1/2 Ln (x 2 +1) + C; as
∫XDX / (X 2 +1) \u003d 1/2 ∫D (x 2 +1) / (x 2 +1) \u003d 1/2 ln (x 2 +1) + c.

Example3.35 . Calculate ∫lnxdx.

Decision. Using the integration formula in parts, we get:
u \u003d lnx, dv \u003d dx, du \u003d 1 / x dx, v \u003d x. Then ∫lnxdx \u003d XLNX - ∫x 1 / x DX \u003d
\u003d XLNX - ∫DX + C \u003d XLNX - X + C.

Example3.36 . Calculate ∫E X SINXDX.

Decision. Denote u \u003d e x, dv \u003d sinxdx, then du \u003d e x dx, v \u003d ∫sinxdx \u003d - cosx → ∫ e x sinxdx \u003d - e x cosx + ∫ e x cosxdx. The integral ∫E X COSXDX also integrate in parts: u \u003d e x, dv \u003d cosxdx, du \u003d e x dx, v \u003d sinx. We have:
∫ E X cosxdx \u003d e x sinx - ∫ e x sinxdx. Received ∫E x sinxdx \u003d - e x cosx + e x sinx - ∫ e x sinxdx, from where 2∫e x sinx dx \u003d - e x cosx + e x sinx + s

Example 3.37. Calculate J \u003d ∫COS (LNX) DX / X.

Decision.Since dx / x \u003d dlnx, then j \u003d ∫COS (LNX) D (LNX). Replacing LNX through T, we come to the table integral J \u003d ∫ Costdt \u003d Sint + C \u003d Sin (LNX) + C.

Example 3.38 . Calculate j \u003d.

Decision. Considering that \u003d D (lnx), we produce the LNX \u003d T substitution. Then J \u003d. .

Primed function and indefinite integral

Fact 1. Integration - action, inverse differentiation, namely, restoring the function according to a known derivative of this function. Function restored F.(x.) Called predo-shaped For function f.(x.).

Definition 1. Function F.(x. f.(x.) at some interval X.if for all values x. equality is performed from this gap F. "(x.)=f.(x.), that is, this feature f.(x.) is a derivative of a primitive function F.(x.). .

For example, a function F.(x.) \u003d SIN. x. is a primary for function f.(x.) \u003d COS. x. on the whole numerical straight, since with any value of the IKSA (sin. x.) "\u003d (COS x.) .

Definition 2. Uncertainly integral function f.(x.) It is called the totality of all its primitive. This uses recording

∫

f.(x.)dX.

,

where sign ∫ called the integral sign, function f.(x.) - a replacement function, and f.(x.)dX. - A concrete expression.

Thus, if F.(x.) - some kind of primary for f.(x.), T.

∫

f.(x.)dX. = F.(x.) +C.

where C. - arbitrary constant (constant).

To understand the meaning of many primitive functions as an indefinite integral, the following analogy is appropriate. Let there be a door (traditional wooden door). Its function is "to be a door." And what is the door made from? From wood. Therefore, a multitude of primitive integrated function "Be the door", that is, it is an indefinite integral, is the function "Being + C", where C is a constant, which in this context may indicate, for example, a tree of wood. Just as the door is made of wood using some tools, the derivative of the "made" function from the primitive function with the formulas that we learned by studying the derivative .

Then the table of the functions of common objects and the corresponding primitive ("to be the door" - "be tree", "be a spoon" - "be metal", etc.) is similar to the table of the main indefinite integrals, which will be shown slightly below. The table of uncertain integrals lists common functions with the indication of the primordial, of which these functions are made. In terms of the tasks to find a indefinite integral, such integrants are given, which without particular gravity can be integrated directly, that is, on the table of uncertain integrals. In the tasks, it is necessary to pre-convert to the tasks to preform so that you can use table integrals.

Fact 2. Restoring the function as a primitive, we must take into account an arbitrary constant (constant) C., so as not to write a list of primitive with different constants from 1 to infinity, you need to record many of the primitive with an arbitrary constant C.For example, as follows: 5 x.³ + p. So, an arbitrary constant (constant) enters the expression of primitive, since the primitive can be a function, for example, 5 x.³ + 4 or 5 x.³ + 3 and with differentiation 4 or 3, or any other constant is applied to zero.

We will put the integration task: for this function f.(x.) find such a function F.(x.), derivative of which equal f.(x.).

Example 1.Find a variety of features

Decision. For this feature, the function is function

Function F.(x.) called primitive for function f.(x.) if derivative F.(x.) Equal f.(x.), or that the same, differential F.(x.) Raven f.(x.) dX..

(2)

Consequently, the function is primitive for a function. However, it is not the only primary for. They also serve as functions

where FROM - Arbitrary constant. This can be seen differentiation.

Thus, if there is one first primary for the function, then it has an infinite multitude of primitive, differing in permanent term. All the primary functions are written in the above form. This follows from the following theorem.

Theorem (formal statement of fact 2).If a F.(x.) - Valid for function f.(x.) at some interval H., then any other primitive for f.(x.) At the same gap can be presented in the form F.(x.) + C.where FROM- Arbitrary constant.

In the following example, we already appeal to the integral table, which will be given in paragraph 3, after the properties of an indefinite integral. We do it before familiarization with the entire table, so that the essence of the foregoing is understood. And after the table and properties we will use them when integrating in all fullness.

Example 2.Find multiple features:

Decision. We find the sets of primitive functions, of which "these functions are made". When mentioning the formulas from the integral table, simply accept that there are such formulas, and we will study the table of uncertain integrals to be completely further.

1) applying formula (7) from the integral table with n. \u003d 3, we get

2) using formula (10) from the integral table with n. \u003d 1/3, we have

3) as

then by formula (7) when n. \u003d -1/4 Find

Under the sign of the integral write not the function itself f. , and her work on differential dX. . This is done primarily in order to indicate which variable is looking for a primitive. For example,

, ;

here, in both cases, the integrand function is equal, but its indefinite integrals in the considered cases are different. In the first case, this feature is considered as a function from a variable x. , and in the second - as a function from z. .

The process of finding an indefinable integral function is called integrating this function.

Geometric meaning of an indefinite integral

Let it be required to find a curve y \u003d f (x) And we already know that the tangent of the tilt angle at each of its point is the specified function f (x) The abscissions of this point.

According to the geometric meaning of the derivative, tangent tilt angle at this point of the curve y \u003d f (x) equal to the value of the derivative F "(x). So you need to find such a function F (x), for which F "(x) \u003d f (x). Function required in the task F (x) is a primary one f (x). The condition of the problem satisfies not one curve, but the family of curves. y \u003d f (x) - one of such curves, and every other curve can be obtained from her parallel transfer along the axis Oy..

Let's call a graph of a primitive function from f (x) integral curve. If a F "(x) \u003d f (x)then the graph of the function y \u003d f (x) There is an integral curve.

Fact 3. An uncertain integral is geometrically represented by the seven of all integrated curves As in the figure below. The remoteness of each curve from the start of coordinates is determined by an arbitrary constant (constant) integration C..

Properties of an indefinite integral

Fact 4. Theorem 1. The derivative of an indefinite integral is equal to the integrand function, and its differential is a source expression.

Fact 5. Theorem 2. Unexposed integral from differential function f.(x.) Equal function f.(x.) with an accuracy of a permanent term .

(3)

Theorems 1 and 2 show that differentiation and integration are mutually reverse operations.

Fact 6. Theorem 3. A constant multiplier in the integrand can be made for a sign of an indefinite integral .

Direct integration using a primitive table (table uncertain integrals)

Predotted table

Find the primary function on a well-known differential function in the event that we use the properties of an uncertain integral. From the table of the main elementary functions using the equality ∫ d f (x) \u003d ∫ f "(x) dx \u003d ∫ f (x) dx \u003d f (x) + C and ∫ k · f (x) dx \u003d k · f (x) DX can be drawn up the table of primitive.

We write the table derivatives in the form of differentials.

Constant y \u003d c C "\u003d 0 The power function y \u003d x p. (x p) "\u003d p · x P - 1	Constant y \u003d c D (C) \u003d 0 · D X Power poundation y \u003d x p. d (x p) \u003d p · x p - 1 · d x
(a x) "\u003d A x · LN A	The indicative function y \u003d a x. d (a x) \u003d a x · ln α · d x In particular, at a \u003d e we have y \u003d e x d (E x) \u003d E x · d x
log a x "\u003d 1 x · LN A	Logarithmic fun y \u003d log a x. D (log a x) \u003d d x x · ln a In particular, at a \u003d e we have y \u003d ln x d (ln x) \u003d d x x
Trigonometric functions. SIN X "\u003d COS X (COS X)" \u003d - SIN X (T G x) "\u003d 1 C O S 2 x (C T G X)" \u003d - 1 SIN 2 x	Trigonometric functions. d sin x \u003d cos x · d x d (cos x) \u003d - sin x · d x d (t g x) \u003d d x c o s 2 x d (c t g x) \u003d - d x sin 2 x
a r c sin x "\u003d 1 1 - x 2 a r c cos x" \u003d - 1 1 - x 2 a r c t g x "\u003d 1 1 + x 2 a r c c t g x" \u003d - 1 1 + x 2	Inverse trigonometric poundations. d a r c sin x \u003d d x 1 - x 2 d a r c cos x \u003d - d x 1 - x 2 d a r c t g x \u003d d x 1 + x 2 d a r c c t g x \u003d - d x 1 + x 2

We illustrate the example described above. Find an indefinite integral of the power function f (x) \u003d x p.

According to the table of differentials D (x p) \u003d p · x p - 1 · d x. According to the properties of an indefinite integral, we have ∫ d (x p) \u003d ∫ p · x p - 1 · d x \u003d p · ∫ x p - 1 · d x \u003d x p + c. Therefore, ∫ xp - 1 · dx \u003d xpp + c p, p ≠ 0. The record version of the record is as follows: ∫ xp · dx \u003d xp + 1 p + 1 + c p + 1 \u003d xp + 1 p + 1 + c 1, P ≠ - 1.

We will take equal to - 1, we find many of the primitive power functions f (x) \u003d x p: ∫ x p · d x \u003d ∫ x - 1 · d x \u003d ∫ d x x.

Now we will need a table of differentials for the natural logarithm D (ln x) \u003d d x x, x\u003e 0, therefore ∫ D (Ln x) \u003d ∫ D x x \u003d ln x. Therefore, ∫ d x x \u003d ln x, x\u003e 0.

Table of primary (uncertain integrals)

In the left column of the table, formulas are placed that are called basic primitive. The right column of formulas are not basic, but can be used when finding uncertain integrals. They can be checked differentiation.

Direct integration

To perform immediate integration, we will use the tables of primitive, integration rules ∫ f (k · x + b) dx \u003d 1 k · f (k · x + b) + c, as well as the properties of uncertain integrals ∫ k · f (x) dx \u003d k · ∫ f (x) dx ∫ (f (x) ± g (x)) dx \u003d ∫ f (x) dx ± ∫ g (x) dx

The table of the main integrals and the properties of the integrals can only be used after easy transformation of the integrand.

Example 1.

We will find the integral ∫ 3 SIN x 2 + COS X 2 2 D X

Decision

We take out from under the sign of the integral coefficient 3:

∫ 3 sin x 2 + cos x 2 2 d x \u003d 3 ∫ sin x 2 + cos x 2 2 d x

According to trigonometry formulas, we convert the integrand function:

3 ∫ sin x 2 + cos x 2 2 dx \u003d 3 ∫ sin x 2 2 + 2 sin x 2 cos x 2 + cos x 2 2 dx \u003d 3 ∫ 1 + 2 sin x 2 cos x 2 dx \u003d 3 ∫ 1 + Sin XDX

Since the integral amount equal to sum Integrals, T.
3 ∫ 1 + sin x d x \u003d 3 ∫ 1 · d x + ∫ sin x d x

We use the data from the table of primitive: 3 ∫ 1 · dx + ∫ sin xdx \u003d 3 (1 · x + C 1 - Cos X + C 2) \u003d \u003d P y with T l 3 s 1 + C 2 \u003d C \u003d 3 x - 3 COS X + C

Answer: ∫ 3 SIN x 2 + COS x 2 2 D x \u003d 3 x - 3 cos x + c.

Example 2.

It is necessary to find many primitive functions f (x) \u003d 2 3 4 x - 7.

Decision

Use the table of primitive for the indicative function: ∫ a x · d x \u003d a x ln a + c. This means that ∫ 2 x · d x \u003d 2 x ln 2 + c.

We use the integration rule ∫ f (k · x + b) d x \u003d 1 k · f (k · x + b) + c.

We obtain ∫ 2 3 4 x - 7 · d x \u003d 1 3 4 · 2 3 4 x - 7 ln 2 + c \u003d 4 3 · 2 3 4 x - 7 ln 2 + c.

Answer: F (x) \u003d 2 3 4 x - 7 \u003d 4 3 · 2 3 4 x - 7 ln 2 + c

Using the table of primitive, properties and the rule of integration, we can find a lot of uncertain integrals. This is possible in cases where the integrand function can be converted.

To find the integral from the logarithm function, the functions of Tangent and Kotangens and a number of others, special methods that we will consider in the section "Basic Integration Methods" are applied.

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