087.5 92

68

ISBN 978-966-14-1305-3 () ISBN 978-5-9910-1585-1 ()

Depositphotos/Oleksiy Fedorov, Iarygin Andrii, Andreus, Elnur Amikishiyev, Wolfgang Filser, K ostyantin Pankin, plrang, , 2011 , , 2011 , , 2011 , . , 2 0 1 1

, . , .

N.N = { 1, 2, 3 ... }

. Z. .

Z = {... -3, -2 , -1, 0, 1, 2, 3 ...}

, / ^ * 0), , . : , , ( ). Q.

Q = J ...-3 , -2 ,-1 , 0, 1, - , 2, 3,... I1 4 8 6 J

, / , , .

I. /2, /, /5, , . ,

(I, ). R. . , .

, . , , 21/2, 2 2 , i ( - i ) . .

, , , , . , .

. z ( , ) z = (; ) , , , . .

z, Rez. z, Im z. : z = Rez + iIm z. : z = + iy, i

, i2 = - 1 . ,

. , ,

, .

4

a + ib = 0 = + i 0 , , .

0 + ib . .

1) a + ib + id , = b = d.

2) a + ib c + id a + c + i(b + d).

3) a + ib c + id a c - b d + i(ad + bc). , ,

. a + ib a - b i

:(a + b i ) ( a - b i ) = a 2 - a b i + a b i - b 2i2 = 2 +b2.

: ib id = i2bd = -bd.

4) a + bi + di 0 , :

a + bi _ ( + bi)(c - di) _ a c + bd b e - a d .c + di (c + d i ) ( c - d i ) c2 + d 2 c2 + d 2

c + di = 0, , .

. z = a + ib , Rez (; >), a Im z (. 1).

, .

. z = a + ib Z (; ) (. 2).

z : |z| = 'Ja2 +b2 . z |z|

. z = a + ib, (z 0)

( z ( -

, ,. 2 ).

= argz. z = 0 .

. : 0 , , .

. b z = a + ib ( z):

(, )

5

a = rcos(p b = rsintp

z = a + ib = r(cos(p + isinip). . x + iy

( I I. , , -

( I 1 0 1: 1= > 1 =

0 1 - 1 0

+ , -

= *

,

=> , ..., ... ,

,

V ,

,

V , ..., ...

3 ,

: = , : , def

/ , ,

1,}, [ 1, 0 , ...

{ 1 , ... , ...

0 ,1 )

6

/ : , ,

, ... ,

,

N

Z ,

Q ,

R ,

; ,

; ;

:

* - : - ...

6

, + , , /

"

( ),

I , : ... ... ... ...

: ... ... ... ...

! :

Jdx : ( ... ...) ... ( d) ...

: ... ...

7

()

: a + b = b + a ; a - b = a + ( - b) : a x b = b * a , =

b

(): (a + b) + c = a + (b + c) : (ab)c = a(bc)

()

: ( + b) = + cb : (ab) =

= = , =

> > , >

\] || , | |

=>= > , =>

,

b , ,

, b , .

=

||

-

, = 1

= , + = +

> , + > +

() () () ,

n - , = ... .

8

:1 . ()" = ". 3. " = "*

2.I " , > .

5. ( ' = ,\6 . " (),

, , .

- , - .

( ), n- , , . . /.

, . n - .

n- . : .

/ = 0 ; VI = 1 . :

, : ,

;

;

, ; '

, ; .

, , () .

, .

(i") * ( \ , -

V

9

(:a , b t i c e R n n n k e N )

" ^ - 2 - 2+ = 2" ^ 2?Ta-2f b - 2f c = 2 c ( > 0, b > 0, > 0)

2n+l[db~c = 2^ i /b 2 2yjabc = \ 2>\ , (abc > 0)

2 [ [= (#0) = 2?[^ , ( > 0, b > 0) 2 ,

1 2"/2"+l/ = ---- ( b * 0) ( ; . )

4 ^ = 2" ^ , (> 0 )

2" ^ = [ " ( 2 ) = 2 7 , ( >0)

2+ 2" = (2- 1>'2"+ 2 = ( ^ ) 2\ ( R)

() () , ().

, , . . .

( ):

1. j = , > , > , ./* ___

: %/"~,

^ ^ : -= = = = = = = = -----------, ( > 0 ).

/ - *

10

A(yja + ylb\ A (y fa + y fb \- > 0 , > 0, .

a - by [ a - y [ b ^-Ja - y f b ^ y f a + ylb^

3. - 7=-----;= - = ----- -----== .

*1 +

yfa + yfb /2 - y/ab + yfb2, y /a - y /b yfa2 + y[ab + yfb2 , ( + ) { ) . -

- l I r i + - ^ + : -j= = = -----= ----------------------------------------------- 1------------

l fa+1/b ( ^ + Vb)(Vfl7 - ^ b + V b r ) +

b , + b 0 .

+ + + ^ + ~=-----1= = ^= ----- / ______ -----f= v = ----------------------------- . b 1 - 4 + ^ + -

, .

A ^ y j a + y / b ^ A ^ y l a + y f b }

- 4 + ^ ~ { - 1 1 + ^ ) { ^ + ~ +

, + * 0 .

^ A ^ f a - y j b ' j A ^y fa -y f l

b

b ^ + i f c b + + t f e b + ) ( & - ~

, .

4. j=---- J .yja - yjb

: ( - ) [ " ~ ' +

+ "~2 + ... + "~2 + "~' ) = " ". x = \fa, y = y[b,

{tfa -yjb^{yla"~' +yla"~2b +...+ yjab"~2 + yjb"~' | = - . :

(1"~' + yja"~2b + ... + yjab"~2 + ylb"~' V=-----= = ---------------------------------------------------- - , ( > 0 , > 0, y ja -y /b a - b

; , b , ).

5. yfa+yfb

( + ){"~ '~

- " ~ 2 + ...+ ( - 2 + (-> ')" ' ) = * + (_ l ) " ^ . = [ , y = y/b,

+ ^ ' -yla"~2b + ... + ( - l ) " 2ylab"~2 + ( - l ) ' yjb"~' | = + (-1 )" 'b . -

^ + 2 ZIb + . . .+ 2< 1 2 + 2 ) : - - = ---------------------------------------------------------------- ,

y /a+ ylb a - b b + b 0.

11

----------------------, > 0 , > ,

-, > , > , > ,

2 + 2?/ + .

6 . j=-----, j=---------------- j=-----j= -j= j= .

Ja+yJb + ylc y la + y lb -y jc / - / - /

, , {yfa + yfb - Vc j >

x(a + b - c - 2 Jab'j. :

{ / + y/b -y /c^{a + b - c - 2 y / a b ^

y/a+y/b+y/c ( + b - )2 - 4ai>

(a + b - c )2 - 4 a b * 0 .

7. - = -----= j= = , ab = c d .yja+yjb+yjc + yld

, , ( \/ + /b j -

-{y fc + y/d'j . :

A ^ y f a + y f b ^ - { y f c + ^ / d ^ A ^ y f a + yfb - / - y fd ^4~a+yTb + y T c + y f d = ( + )2_( + ) 2

a > 0 , b > 0 , > 0 , d > 0 , a + b ^ c + d .

8. -j= -j= -j=yja +yjb+yjc

(x + y + z ) (x2 + 2 + z 2 - x y - x z - y z ) = x* + 1 + z 3 -3 x y z . -

= /, y = yfb, z = y/c,-ro{yfa + y fb + y fc ^ \ fa 2 + yfb2 + + yfc2 - ^ fa b -y fa c - y fb c ^ j =

= a + b + c -3y/abc.

= ( + b + )2 + 3( + b + c)yfabc + 9yj(abc)2,

( + b + - /abcJ = (a + b + )3 - 27abc.

2 + y f ^ + /2 - y / a b - / - y /b c Y : ----------------------------------------------------------- ,

( + b + ) - 27abc yfa + yfb + yfc 0 , (a + b + c)3 .

/ / (). :

V a 3 I = + / - / - 2 V 2

> , > 0 2 - > 0. , 2 - (

, - , s, = s2).

12

b , , . log0 . :

log, b = x : ' = b ; log, 1 = 0 ( = 1 ); loga = 1 ( 1 = )

> 0, * 0. :

log, (be) = logn i) + log |c |, (be > 0 )

log,, j = log |b| - log,, |c |, ^ > 0 j lg(b ', ) = p lo g 0 |b |, (bp > 0 ) ;

: log > =m

,

lg . b = 7 logb

, , ,

1 , = - i o g a b

lg. b = lo g a b

logc

1g a

13

-4 -3 - 2 - 1 0

-2-3-4

1 2 3 4

() ( ) ( ) , , z ( , , ) (. 3,4).

. . , OY

, OZ . .

, . .

( ) ( , () 1), . , .

i , j , k , , .

, - (. 3, 4).

(I, II, III, IV), OY, .

: ( ; ; /

, , , .

. 3

( ; , )

e y

. 4

, (. 5).

() , ( ).

. , . ,

( , , ).

: (; + 2) N . ' 5

= 0, . , . R ,

() = = , 0 < - < 2 (. 6 ). -

1 2 - / ( ) ] 2

14

. .

(; ) , (; 0,2

= , = 0 ,

15

f( .x) = g(x), / g .

() , .

, . , .

, ( ) .

: + = 0. . = = 0, ,

Vo : 0 = 0. = 0, 0, , 0 a e R : a O = - b ^ O . * 0, 0, -V = -/.

2 + + = 0, 0.'

, , 0 2 D = b2 - 4 .

D > 0 - 4 - 4 , , = -------------------- ;

2 D = 0

= ; (, , 2

), 2

D < 0 ; , :

- b \ l b 2 -4 , ~ 2

, :

- 1 \J-b2 + 4

'- ~ 2

+ + q = 0 , = 1 ( )

16

2 +2 + = 0 , D - 4(2 - ) D > 0:

- - \ 1 2 - - + \12 - ,= , 2 = ;

D = 0: , = - /

2 + = 0, * 0 II 0 * II 1 >

2 + = 0, < 0*1 = ~ 4 ~ ! ' *2 = yj~C/ a

2 = 0 .= 0

( )

,, 2 + + = 0, : x t + x 2 = ,

, 2 = - .

2 + + = 0 : , + 2 = -b , , 2 = .

D > 0 : 1 + + = ( - , ) ( - 2) .

D = 0 : 2 + + = ( - ,)2.

4 + 2 + = 0 2 = .

* =, > 0, 1. > 0 = log0 b. < 0 .

log = b, > 0, 1.

= .

= - 1 =0 = 1sin* = ,

- 1 = (--1) arcsina + = - / 2 + 2 = = /2 + 2

17

= - 1 = 0 = 1

cosx = a,|| < 1

= ia rc c o sa + 2 = + 2 = /2 + = 2

tg j = , (-; )

= arctg + 7 = - / 4 + = = /4 +

ctg = , (-; )

= arcctga + = - / 4 + = /2 + = /4 +

,

|/( ) | = , < 0, = 0, / 0 0 = 0 .

> 0, :

.

' / 0 0 = / ( ) = -

|/M | = *W (

I

:7 * 0 0 s o .

1 / ( * ) = *(*);( * 0 0 > 0 ,

1 - m = g (X)4

f / W > 0 ,

j / ( x ) = * 0 0 ;

j / ( x ) > 0 ,

| - / ( * ) = *(*)

|/(*)|=|*| f ( x ) = g 2(x)

'/ ( * )= * ( * ) . f ( x ) = -g (x ) .

, .

=

=

18

= = \1 )

= 2 + + = yjax2 +bx + c

- y = Pn(x) /Q m(x), () Qm(x) - - m -

{ ( ) g(x) X, 2. f ( x ) < g(x). ( < >, .)

() , .

, / ( ) < g(a).

, .

. . . X : / () < g ( x ) ,

f ( x ) < g l(x), , f ( x ) < g(x) / < 8 1 (*)> : , - , .

.

= const, > 0 ; < 0 , .

f ( x ) g(x) > g(x) + h(x) .

h (x )> 0 X, f ( x ) > g(x) f ( x ) h ( x ) > g(x )h (x ) .

h ( x )< 0 X, f ( x ) > g(x) f ( x ) h ( x ) < g(x )h (x ) .

f ( x ) > 0 , g(x) > 0 X, / () > g(x) f 2(x) > g 2(x) .

19

( - , )' ( - ... ( - )"* < 0 . ( < >, .)

. , 2 + + .

D = b2 - 4 .

D < 0 D > 0

II

> 0 ,2 + + > 0

(00; ) ( - ; x , ] U [ x 2;oo) bX = -----2

> 0 ,2 + + > 0

X (-; ) x e ( - c c ; x , ] U [ x 2; )

< 0 ,2 + + > 0

[ . ; 2 ] 1IIX

< 0 ,2 + + > 0

( , ; 2)

( 0 ,

/ ( ) < 2().

^ / () > () : J (p (x )> 0 , () < 0 ,

| / ( ) > 2"(); | / ( ) > 0 .

= (). R :

| |> 0 ; || = 0 |^ | | - - .

20

.

= sinx, = cosx = 2, s in x > cos > ( > (|| < 1) e (arcsina + 2rai, J i-a rc s in a + 27tn), V Z

sinx < ( | |< 1) ( - - arcsine + 2nn, arcsina + 2nn), V n e Z

cosx > ( | |< 1) x e (-arccos a + 2 nn, arccosa + 2nn), V n e Z

cos < (|| < 1) x e (arccosa + 2 , 2n - arccosa + 2nn), V n e Z

tg x > a x e (arctga + nn, n/2 + nn), \ / n e Z

tg x < x e ( - n / 2 + nn,arctga + nn), V n e Z

ctgx > xe (n n ,a rc c tg a + nn), V n e Z

ctgx < x e (arcctga + nn, n + nn), V n e Z

, .

: f ( x ) = , , ; , , .

, : X; ; () , ,

. ,

() .

, , , . , , : f ( x ) = .

21

, , , . , , / ( ) = ' , 1, = 0,5.

. ,

, .

f (x) > 0 (, ), / () .

f ' ( x ) < 0 (, ), / () .

: , , , .

.

.

/, /, ( 0) / + , , , 0 6 > 0 , | - \ < 8 |/ ( ) - / ( ) | < .

, / , : lim / ( ) = /( ) .-*

/ , .

/ ( ) (. . , ).

/ ( ) , / ( ) = , , .

/ ( ) , lim ^/(x ) lim ^/(x ).

.1. : lim ^ /(x )= lim ^/(x ).

.

22

/(- 0 - +0 |

f ( x ) = , .

, : f ( - x ) = f { x ) .

(. 8 ).

, : f ( - x ) = - f ( x ) .

(. 9).

, .

, (

), .

: , .

y = f ( x ) , 0, f x i}, , f ( x ) > / ( 0).

, , .

, / \ ) < 0 < 0 f (x) > 0 > 0< . . , 0 ( . 10 ( - /2 + 2; - 1), Z ).

= f ( x ) 0 , , , 0, . . 0, 0, , f i x ) < f ( x 0).

, , .

, / '( * ) > 0 < 0 / '( * ) < 0 > 0, . . 0,

. 9

23

= sinx 7 1

\ - /2 / \ \ /2 22 - /2 N. | 0 /2 V ! / \ . 1 /

-1

. 10

0 ( . 10 (/ 2 + 2; 1 ) , k e Z ).

, 0 .

0 () (), () .

. = f i x ) = 0 , .

. .

, 0, (, , 0). , = 0 . 0 , .

/ 0> f ' ( x Q) = 0, / " ( 0)* 0 . /% 0 )< 0 , 0 . f (xQ)> 0, 0 . / 0

/'(*)= * )= = " (*0)= , /'" (*) * . ^ ,) (0) 0, 0 .

/ ()(0 )> 0 , . , 0 .

() f i x ) ( ) (; b), , (; ),

] ; ^ (. 1 1 ).

( ) (. 12 ).1. / ' ] ; [ , / ]; b[

( / ' , / ).2. f (x) > 0 \ / ] ; b[, / .

3. / '( ) < 0 V x Jfl; b [, / .

. = / ( ) 0, 0 . 0 / , 0 /, (0; 0) / . 12

.

1

7 ^

y = f W

' 0 X

24

, (jc0; ,,), . / / ' ( . 13 0 (0 ; 0 ) / ( ) = 3).

: / ' ( , ) = .

:1. /"(*) 0, 0 .2. f (x0) = f m(x0 ) = ... = (0) = 0, / (| (0) # 0, 0 , 0 .

, f ( x ) < , / (. . 14; , (=; 0 ) = 1 / ).

, f ( x ) < b , / (. . 15; , , = 2 ).

, , = sinx (. 10 ).

, , (. . 14; ( - ; )

= 3 ).

. 14

= f i x ) , , .

= :1. l i m / ( x ) = ao;

X-KJ-02 . lim f ( x ) = .

>d+0

= / = 0 > (. 16).

= lim f ( x ) = a.

25

= * = > - (. 17).

= + :

1. h m ^ Ui-> 2. lim ( / ( ) - ) = b.

JC (

), > + ( > - ) .X 1

/ ( * ) = + - = = / 2 (. 18).2 \/

f ( x ) , , , : / ( + ) = /( ) . . .

= f ( x ) ( ) ( ; ) , (; d), (; d) = g(y) (;), = / ( ) .

>> = / ( ) . / ( ) = 0 . , , ,

. g / , /

g, = / ( ) x = g(y) , . . , = / ( ) , = g(y).

y = f ( x ) x = g (y ): f n g . , = / ( )

= g(y) . , f ( g ( y ) ) = y g ( f ( x ) ) = x.

26

, y = f ( x ) = g(y ) .

= . = f ( x ) , x = g(y ) ; y = f ( x ) ,

= g(y) . = = ". = = logo .

.

, (. . ; F G G F ) ; , , , , .

, , . .

, .

. ( ).

, , , , , , .

( ) ( ).

, . , ,

, , .

1- : + = , .

(. 19). 0 -/ .

/ .

27

= ( ).

: , (. ),

1. : D (y) = (; + 0, R ( > 0 ); = , < 0 , R ( < 0 ); = , = 0, R ( - 0 ).

= ( 1, 2, , , . . . > )

f ( x ) = a0 + alx l +a2x 1 + ... + at:x n, 0, , , 2..... .

- ,, 2, 3..... , .

0 = 0 , . , , 2, 3,..., 0, , , 2.....

, ( + 1) - ,, 2, 3 ..... - = 0 + ,, + 2 +... + , = 1 .

. = ,

( ). ,

, (. 2 0 ).1. : D ( /) = (-; + ), . .

R.2. : * 0 : () = (-; + ), . . R; = 0 : ( ) = 0 .

28

3. () : = , . . / ( - ) = ( - ) = - = - / ().

4. : * 0 , = 0 = 0 ; = 0 , = 0 .5. : > 0 , ; < 0 , .

= / , , 0.

(. 2 1 ).

, , .1. : D(y) = (*>; 0) U (0; + ).

7 2. : () = (-; 0) U (0 ;-).3. () :

, . . / ( - ) = = = - / ( ) .

X X4. .5. = 0 -> +.6 . = 0 , . . / ( ) >+ - 0 + / ( * ) - > - - > 0 - .

7. : < 0 > 0 , - = 0 .

. 22

= 2 + + , , , , .

, = = 0, = 2.

= 2 , . (. 22).

.

= 2 + + (. 23).

: 2

= ----- , = ------ .' 2 / 4

: D = b2 - 4.

29

, , ( ) -2 . , - + 4 - - 4

+ + = 0. : , = ------------; 2 = ------------ .2 2

. 23, 24.1. :

D(y) = (-; + ), . . R.2. :

() = \_~D/4 , + ), > 0 ;

(y) = ( -o o ;-D /4 a ] , < 0 .

3. () : = = 0 .

4. : D < 0 ; D = 0 = -/2; D >

- 4 12= ------------.

25. : a > 0 ( a < 0 ) (; /2 ] ;

> 0 ( < 0 ) [ - /2 ;+ ).6 . : > 0 ( < 0 ) / ( x ) min = -D /4 a .

\ , R.

s N - (. 25). N = "

- (. 26). = 1 (. 25) (.

): = . = 2 (. 25) (.

): = 2. = -1 (. 26)

(. ): = / . , . = 0 (. 25)

: = . , X, , . . 0 .

= 2, a e N , . . (. 27) = 1.1. : D ( /) = .2. : ( / ) = [ 0 ; + |\

3. () : .4. = 0.5. :

J ; ] [ 0 ; +

30

6 . : = 0. = 2 - 1, N , . .

(. 28): = 2"-'.1. : D ( f ) = .2. : ( / ) = .3. () : .4. = 0.5. : .

= -2 , N , . . (. 29) = i j x 2".1. : D ( f ) = R \ j 0 }.

2. : ( / ) = ]0; + |\3. () : .4. .5. = 0,

> +.6 . = 0 , . . f ( x ) > - 0 + f ( x ) > -

-> 0 - .7. : ] * ; [

] 0 ; +|\

= - (2 -1 ), N , . . (. 30): = l / 2~1.1. : D ( f ) = R \ {}.2. : ( / ) = R\{o}.

. 26

,

1 = \\ \

\ \ 11

\ 1 1

* ^ -1 0 1 *

. 27

,

-1

75 -1

. 30

31

3. () : .4. .5. 0 .6 . = 0 , . . / ( ) - + 0 + /(jc ) >

> 0 .7. : ]-; [ ]0; + |\

n e N : = ".1. : D ( f ) = ]0; + | .

2. : ( / ) = ] 0 ; + |\

D ( f ) ( / ) .

f ( x ) = ax, = ( = 2,7182818284590452...)(. 31).

= = ().1. : D ( /) = .2. : ( / ) = ] 0 ; + [\3. : .4. . 0 5. = 0 . 31

-> - .

= .

, , : = mjn ( Z, N ). = 1

: > 0 , = y fa " ; = 0 , = 1, , X, . . ;

< 0 , = ( > 0 ). 1*1

1, : = 1| .1. : D ( /) = .

2. : ( / ) = ]0; + |\3. : 0 < < 1

, > 1 (. 32). . 324. .5. 0 < < 1 - 0 >-.

> 1 = 0 > -.

f ( x ) = loga . > 0, * 1, > 0.

: D (y) = ]0; + o[,

() = ] ; + |\ .

32

. 33

1 - -

= log2 .

= i0g ' 9 = log0 ,

> 0, * 1 (. 33).

1. : D ( f ) = ]0; + [.^ 2. : ( / ) = .

3. : 0 < < 1 , > 1 .

4. = !, . . log01 = 0.5. 0 < < 1 > 1 = 0 - 0.

= \ (. 34).1. : D ( / ) = ]0 ;+ [.2. : ( / ) = .3. : .4. = 1, . . 11 = 0.5. = 0 .. 34

= sinx (. 35).1. : D ( f ) = .2. : ( / ) = ] - l ; l [ .3. () : .4. : = 2.5. : [ - /2 + 2; /2 + 2 ] ,

Z , , [ /2 + 2\ /2 + 2:], Z , .6 . : (/2 + 2-, 1),

( - /2 + 2-,-1), k e Z .7. = , Z.

= cosx (. 36).1. : D ( f ) = .

33

2. : ( / ) = ] - l ; l [ .3. () : .4. : = 2.5. : [2 -, + 2 J, k e Z \

: [ - + 2:;2:], Z.6 . : (2; 1), k e Z ;

( + 2;-1), k s Z .7. = /2 + , e Z .

y = tg x (. 37).

1. : D ( f ) = R \ {/2 + \ Z j .2. : ( / ) = R.3. () :

.4. : = .5. :

] - / 2 + ; /2 + [, fceZ .

6 . = , k e Z .

7. = /2 + , k e Z , , . .

lim tg(x) = -oo, lim tg(x) = +-x - > + n-Jt+0 - - + - 0

2 2

8 . , . = (2 -1 ) /2 , Z.

= ctgx (. 38).1. : D ( f ) = R \{/ | e Z j .2. : ( / ) = R.3. () :

.4. : = .5. :

[; + [, e Z.

6 . = /2 + , Z.

7. = , Z , , . . jg

lim tg(x) = +oo, lim tg(x) = -oo. - ^ -fc+O x - m - fc -0

8 . , . = , k e Z .

34

()

= arcsinx (. 39).1. : D ( / ) = [ - l ; l J .

2. : ( / ) = [ - /2 ; /2 ] .3. () : .4. : .5. = 0.

y = arccos* (. 40).

1. : D ( /) = [ - l ; l ] .

2. : ( / ) = [0; ] .3. () : , .4. : .

. 39 5. = 1.

= arctg (. 41).1. : D ( /) = .

2. : ( / ) = J-T t/2; /2 [ .3. () : .4. : .5. = - n / l - /2

>.6 . = 0.

35

arcctgx. (. 42).1. : D ( f ) = .2. : ( / ) = ]0 ; [ .3. () : , .4. : .5. = > - = 0

X > + 0 0 .6 . .

, .

- ~ = shx = ---- -----

1. : D ( /) = .2. : ( / ) = .3. () :

.4. : .6 . = 0.

= ch =

(. 43).1. : D ( /) = .

2. : ( / ) = [ l; + [.3. () :

.4. :

] - ;[ ] 0 ;+ [\5. : (0; 1)

.6 . .

shx

= th = ----- =chx

~ (. 44).

+ '1. : D ( f ) = .2. : ( / ) = ] - ! ; l [ .

. 43

36

3. () : .4. : .5 . : - - 1 ( > - ) = 1 ( + ).6 . = 0.

chx +~ ,

= cth x = ----- = -------- (. 44).shx -

1. : D ( f ) = R \ {}.

2. : E ( f ) = R \ ] - ] ; l [ .3. () : .4. : ] - ; 0 [ ]0 ; + [ .5 . : = - 1 ( > - ) = 1 ( > + ) .

: = 0 0.6 . .

. , . , = / ( ) , N, N ( ), = /{ ) 1, 2, . . . ,,....

, , , = + . .

( ) , ( ) : { < 2 < 3 < ...... 2 > 3 >...

, , , d, . d () . an = , + ( n - l ) d .

, , , q , . q -

37

. bn = b, q"~'.

, .

/ ( ) = / () = 0, (, ) , / '

, / W - / ( ) ( > (, ), - - ------= / ()

(a',b) N : = lim e > - 00.

.

.

lim = , .* ( ) ( ) , lim(a ) = lim a + lim ;

- - -

1 lim(ca ) = c lim e , ; lim (a b ) = lim a . l i m bn ; 1 - ^ = "^* , -+ .-*00 n->D -5 -0 - fo lim p

38

lim an = < bn Vn, < .-

0 8 ( e ) > 0 , < Ijc | < 5 / ( - ) < . : lim x

1 1 ' I - / ( ) - .

/ ( ) - , / ( - 0 ) = / ( + 0 ).

.

lim = , .-* - > / ( ) , g(x),

lim (/(x ) g(x)) = lim f ( x ) lim g ( x ) ; lim (/(x ) g(x)) = l im /(x ) l im g (x ) ;x-*a x *a x *a x *ti z *a x-*a

n x ) l im /(x )lim ------= 2------- , l im g (x )* 0 .*- g(x) lim g(x) *-

x-*a

.

, .

: / , / ,0 , , 0 , , 1 .

) -

() ( 0/0 /), - ().

(): :

Um lim () , , VI/'()

= , , , ( = 0 , ) , (), ( ) , V|/'(x) 0 ;

'(), /'() ( ). / /

, . .

: - ; 0 ; 0, , 1" , - : 0/0 /, .

/ ( ) =

39

:

()() = ( ): )

/ ( ) = ()\|/(), > ( > ) ( - ),

() = ------ ; V'(-) ------- , 0 /0 :() v M

v (x )-u (x )f i x ) = ---------------.' m ( x ) v ( x )

f ( x ) = ( )

40

* 3 * 5 ! -1 X2" sinx = x ------- h------... + ( - 1) --------------I-..., x e R ;3! 5! (2m -1)!2 4 2

cosx = l ------+ -------... + ( - 1) -------- + ..., ;2! 4! (2)!

3 25 177 629 te x = x + + ------ -------+ ----------- < < ;

3 3-5 3 *5-7 3 5 7 9 2 21 1-3 5 1-3-5 7

arcsinx = x-l------ + --------- + ------------ X ;2-3 2 -4 -5 2 -4 -6 -73 5 2*i-l

arctgx = + ... + ( I)" 1 +..., l ; l l .3 5 2 - 1 L J

1 ( 1 )2 ( 1 )3 = 1------ +-------------------- 1- ..., < < ;

1! 2 ! 3!, 2 3

= 1-1----1-----1-----1-..., 0 , * 1 ; lim ----------------= 1 . 1 *-* olx

41

, :

/ '( * ) = / ( * + / ( ) .-0

., ,

. .

. , , . .

, / .

(cf)' = c f

( f + g ) ' - = f '+ g '

( f - g ) ' = f ' - g '

/ g

( f g ) ' = f ' g + f g '

, ( f * j =(e*l" ' j +

/ > 0

' = 0 .

: ' = 1.

: = ; (/*) =

: (sinjc)' = cosx; (cosx)' = - s in x ; (tgx)' = ; (ctgx)' = ------ .cos x sin x

: (e*) - l1 ; (a ' j = a* ln a .

: ( ln x ) = ; (log x ) = p .x x ln a

42

: = (), : = ' ' ; (sin )' = cosh ; (cosu)' = - s in u - ';

(tgu) = ;(c tgu )' = -----; ( 7 ) = - ^ = ; ( ) = -;() = 1 ; ( 1 ) = ;cos sin 2yju

u ln a

, . ( ).

F / I, : F'{x) = f ( . x ) V x e l .

/ / F(x) + , .

F / , G g,

F + G f + g : F ' = f G '= g, (F+ G)'= F '+ G '= f + g. F /, ,

kF kf. (kF )' = kF' = kf. (k F ) ' = kF = kf. F(x) f ( x ) , ,

* 0 , F(kx + b) f ( k x + b) :

^ F ( f a c + b ) j = j F ' ( k x + b ) x k = f ( k x + b).

/ / .

/ ( 0 , 1 , 0) .

.

. [; b] , / , , [a; b] = = (. 45), .

/ [; fcj, F , S [; b j , . . S = F(b)-F(a).

43

[a; b] S (. 46):

s = ^ ( / ( * ) + / ( * , ) + + '/(* -,))

> 0 -> S (. 47). -

b J > Sn > j f ( x )d x ,

;/ , .

, f ( x ) .

1

. 47

f ( x ) (;) F(x)

, . . F'(x) = f ( x ) < < , j f ( x ) d x = F'(x) + C, < < , .

(J /(x )< ix ) = / ( ) , jF '{x)dx = F(.x) + C,

d j f ( x ) d x = f ( x ) d x ; jdF(x) = F( x) + ;

j f ( x ) d x = F(x) + C, to j f ( a x + b)dx = F(ax + b) + C, a f 0;

f ( a f ( x ) + f}g(x))dx = a j f ( x ) d x + p f g ( x ) dx , a 2 + p 2 * 0

du = d(u + C); du = - d ( a u ) ; du = - d ( u 2) ; cosu du = d{sinu) ; n ? v /

i u d u = -d (c o s u ); = d(ln|w|); f ' ( u ) - d u = d( f (u ) )

jg {x )dx = G(x) + C, jg (u)du = G{u) + , =

44

g(x) = g l(.x) + g 2(x), jg (x )d x = j g l(x)dx+ j g 2(x)dx

g(x) , , =

45

rea dx 1 f ea (ea d x \-------= ----------------- + ------ n 1

J x" n - l ^ x"-' J x -' J

je Inxdx = - e a In|jc - Ei(cx), Ei(cx)

f e e sin b xdx = ----- -(cs inbx -b co sb x )3 +bf\ea cos b x d x - ----- -(cosbx + bsinbx)J +bf ex . e^sin"-1* , . 4 r t ( r t - l ) f . n_2 ,\e sin x d x = :----- : (csm x - ncosx) + ------ \e sin xdxJ +n +n Je . cos"-1 x . . . n(n - 1) r _2 ,e cos x d x - :----- - (ccosx + Hsmx) + :------ \e cos xdx

J + n +n J

{xedx = eJ 7r

f dx = i ( 1 + erf * j t ), erf(...) aV 2 Tt 2 ^ a v 2 j

^ 2 rx ir~l ) 2n J x in

fsincxcfct = - - c o s cx J f . , sin"'1 cxcoscx - 1 . -2 ,sin cxdx = --------------------- + ------ sin cxdx, n> 0J nc n Jr , sincx xcoscxxsin cxdx = -----------------

J , . 2 coscx I x s in cx x 2coscx\x" sin cxdx = ----------- h----- ---------------------J

6 sincc 6xcos cx 3x2 sincx x3coscxf 3 . , osincx \x sincx ax = ------------- hJ c4f 4 . 24 cc\x sincxax = -------- ;J c

24 cos cx 24x sin cx 12x2 cos cx 4x 3 sin cx x 4 cos cx

------- "120 sin cx 120xcos cx 60x2 sincx 20xJ coscx 5x4 sincx

x""1 x"-3 x"~5

2 -(-1)! 4 ( -3 )! 6 -( -5 )!

- 4

Jx 5* in m fe = ^

Jx" sin cxdx = n ls in cx

- u lc o s c x , i .c-nl ( -2 ) ! - ( -4 )!

fx" sincxdx = coscx + fx_l cos cxdx, n> 0 J 1

& = (- 1 ()1 J U (2 / + 1) -(2 i + 1)!rsincx , sincx f cos cx .I------- dx = --------------- r + ------I dx

x 5 coscx

46

t dx 1, J-----= "ln tgJ sincx 2 dx _ coscx'sin" cx c(l - n)sin"_1 cx

r dx _ 1 f cx _ l i s i n c c c g U + 4r x d x x f cx 4! 2 .------------= - t g --------- + I" In

h + sincx 1^2 4 J r xd x x ( n ex') 2 ,---------= - c t g -------

4 sincx 1 4 2 J f sin cxdx l ( cx ----------- = X + - t g - +

J l sin c x I 4 2

n - 2 r dx-------- ^ n - 1 J sin cx

>1

7U c xsin | --------

4 2

lsinc.xsinc.xdx =sin(c, - c2)x sin(c, +c2)x

2 (Cj - c 2) 2 (c, + c2)

cos cxdx = -sincx J

cxsincx n - 1----------+-----nc n

co s cx X s in cx x c o s cxdx = - -----------

jcos" 2 cxdx, n> 0

x sincx n fx 1 sin LX fix

J

Jcos cxdx =

Jjx" cos cxdx

(E S l f ld x = In Icxl + ( - 1)'J x 1 1 t t 2i-(2i)\fc o sc x . c o s c x fs in c x ,I--------dx = --------------- ;------------------- d x , n * l* x ( n - l ) x f l - l - x

I;

(cx)1'

( - D x - ( cx n+1

X

dxcoscx

dx

Jcosn cx c ( n - l)cosn cxr dx 1 cxf------------= - t g J l + coscx 2r dx 1 cxJ------------ = ctgJl- c o s c x 2r x d x X cx 2 ,------------ = - t g + In

J l +coscx 2 r x d x x cx 2 ,------------= ctg + In

J 1 -c o sc x 2 c cos cxdx l cx------------ x tg

J l + coscx 2rcoscxdx l cxj------------ = - x ctg' I - coscx 2

. i z l f_n - 1

dx n>l

r sin(c, - c 2)x s in (c ,+ c2)x , , , ,cosc.xcosc.xax = ------ - 1 -------- ------ , c, * c,

J 1 2 2 ( c , - c 2) 2 (c,+c2) 1,1 121

47

tg c x - 1 2 2

seccxdx = iln lseccx + tgcxl ' 1

tg cxdx = In|coscx|

tgcxdx = - tg'c x - ftg"~2cxdx, 1 c ( n - l ) 1

dx X 1 . I . I- = + In sincx + coscx

tgcx + 1 2 2c dx

---------- = ------ 1---- ln|sm cx -coscxltg cx - 1 2 2ctg cxdx

ln|sin

c X l . i . I- = -------- in sin cx + cos cx1 2 2c 1 1

tgcxdx x 1 , 1 - I- + ln |sincx-coscx l

tgcx +

sec" cxsincx n - 2 rsec cxdx = -------------------- 1- ------ sec cxdx, n * 1

( 1) 1dx x--I7 = x_tgTsecx + 1 2

coseccxdx = --ln lco seccx + ctgcxl

cosec"- 'cxcoscx n - 2 r .cosec cxdx = -------------------------- 1--------cosec cxdx, n * l

' " 1 ( -1 )

ctgcxdx = iln |s in cx |

ctg"cxdx = ------ - ctg" 'cx - fctg" 2cxdx, n * 1c(n-l) 3c { n - 1)dx _ f tg cxdx

1 + ctgcx ^tgcx + 1dx _ rtg cxdx

1 - c tg c x ^ tg cx - 1

dx 1 .=ln

coscx sincx ~ c S, cx ,

d x 1 7 tr = tg CX + -

(coscx sincx) 2c ^ 4

_____ = _ ! _ ( s in x - c o s x _ 2( _ 2 ) f______ ^ ______(cosx + sinx)" H - l l^ c o s x + s in x )" '1 (cosx + sinx)" 2

+ Ini sir7 T r I

cos cxdxI sincx + coscx I

coscx + sincx 2 2 c

cos cxdx x 1 I I- = ---------In s in c x -co scx

co scx -s in cx 2 2c

sin cxdx

coscx + sincx 2 2clnlsi T/- Isincx + coscx

48

sin cxdx x 1 , | .---------------= ------------ln ls in c x -c

2 2c 1co scx -s in cx cos cxdx 1 1 ,

----- tg + Insincx(l +coscx) 4c 2 2c

cos cxdx l 2 cx 1 ,------------------= ------ ctg2 ---------- In

sincx (l-co scx ) 4c 2 2c

cxtgT

cxt67

sin cxdx cx ] 1 . + - + In 2 4 J 2 c

, CX 71

tgl T + 4cx + 2 4

-In2c

, cx t g l T + -

coscx(l +sincx) 4c sin cxdx l

coscx (l-sincx ) 4c g

sincx coscx dx = sin2 cx 2c

cos(c, + c2 )x cos(c, - c2 )xsinc,xcosc2xdx =

sin cx cos cxdx =

sin cxcos" cxdx =

sin cxcos cxdx

2 (c. + c2) 2 (ci - c , )1

-s in cx, n * 1c(n + l)

1

c(n + l)

sin""1 cxcos' cx n - 1 c(n + m) n + m

sin" excos" cxdx =sin cx cos

c(n + m)dx 1 , I |

------------= - l n tgcxsin or coscx 1

dx 1

cx m - 1 + -------

dx

jsin" 2 cxcos cxdx

jsin" cxcos -2 cxdx;

b 1sincxcos"cx ( 1) cos"-1 cx sincxcos""2 cx dx 1 r dx

sin"cxcoscx c (n -l) s in " 1 cx 's in '' cxcoscx sincxdx l

, n * 1

, n * 1

cos cx c (n -l)c o s" cxsin2 cxdx l l

---------- = sincx + - lncoscx

sin2 cxdx sincx

, * 1

cos' cx c ( n - l)cos_1 cx :in" cxdx

. cx ~4+ ~2

Jn 1 dx

n * 1

sin" 1 cx fSin" 2 cxdx+ --------------- , n * 1

J rn s r rc ( n - l )

:in cxdx

cos cx c (m - l)c o s cx

,sin" cxdx sin"-1 cx

n - m + 2 fSin cxdx---------- > mm - 1 3 cos cx

n - 1 fsin""2 cxdx -------------------------- , m * n m JJ cosmcx c(H -m )cos cx n ~ m i cosm cx

t s\n" cxdx sin-1 cx

cosmcx c (m - l)c o s m cx m

n - 1 rs in -1 c x d x~ > m * 1 1 J cos cx

; m ,n > 0

m,n> 0

49

, cos cxdx iI------------ ------------------ 13 s in " cx c(n l ) s in " cx

o s 2 cxdx;----------= - | COSCX + Ins in c x c ^

rco s2 cxdx

cxt g y

pCOs c x d x l { c o sc x r dx )f ; = -------- t j + f-------- 5 > * 1J sin cx - l ^ c s in cx J sin cx J /cos"cxdx cos"+1 cx - - 2 i-cos"cxdx

* sin" c x c ( m - 1)sin"'-1 cx m 1 *sin2cx 1

|.cos cxdx ei

cos" 1 cx n - l rc o s n cxdx + ------- ------- --------, m * n

c i r isin cx c(n - m)sin' cx n - m 3 sin cx

cos"-1 cx------------- : I - .sin cx

/cos" cxdx

^ s in 1 cx c(n-l)sin '"'c x m

Jsin cx tgcx dx = - ( In | sec cx + tg cx | - sin cx )

rtg "cxdx lp ;-----= -----------tg" 'cx, n I sin cx c ( n - l )

ftg"cxdx i

n - 1 fCos" 2 cxdx- f------ ----- , m 1

- 1 J s in

J cos1 cx c(n + 1) rdg"cxdx i

tg cx , - 1

ctg"*'cx, - \3 s in 2 cx c (n + l)

fc t e"cxdx ij - 1 "!----- = T77LTTtg ,"', ' * 1cos2 cx c ( l - n )

tgm (cx)p u dx = - 1 - tg"1*'1-1 (cx) - J tg ^ dx n4t lt " r t a l i - v lctg "(cx) c(m + n - l ) 3 ctg(cx)

x > 0Jlncxdx = xlncx - X

|( ln x )2dx = x(lnx)2 - 2 x ln x + 2x

j(lncx)"dx = x(lncx)" - J(lncx)"_1dx

f = ln |lnx | + y ^ ln x 1 1 t t >'!

dx x l r dxJ;(lnx)" (n - l) ( ln x ) ' Jx " ln x d x = x"'1

* ' 7 1

m + 1 (m + 1)

"" ( ln x )"

(lnx)"

m - 1

n 1

fx"(Inx)"dx = - ------ --------- fx (lnx)" 'dx, - 13 m + 1 m + 1 3

(lnx )V x (lnx)"*'X

In xdx + 1

lnx

- l

1

( r n - l ) x m-' ( m - l ) 2x

50

(In x)"dx

xx mdx

(lnx)"

( m - l ) x n 1 3(lnx)" 'dx

m -m + 1 r x mdx

, 1

(lnx)"

x mdx lnx

- ^ - = ln|L x ln x

dx

m + 1 r:r r+"^T Ji(n 1)(1)" n - 1 (lnx)"

= Ei((m + l)lnx), Ei(x)

lnx

x " ln xdx

= lnjl]n x |+ ( - ! ) ' i = l 1

( (w - l) '( ln x )1

i i!

* 1x(lnx)" ( 1)(1)"

xsin(lnx)rfx = (sin (lnx) - cos (lnx))

2 x

cos (In x )d x = (sin (In x) + cos (In x))

-

- d x = -a ( l - n ) x - b

( + b)" ( -1 )( -2 )( b) x2 ^ \ ( (ax + b)7

ax + b a3 2

- {1.2}

- 2b(ax + b) + b2 ln\ax + b\

(ax + b)2dx = - ^ A a x + b - 2bln|ox + b\ -

(ax + b)

- dx = ^ \ (ax + b)" a1 '

{1, 2, 3}

dx 1 .-------- = In

x(ax + b) b

dx

j d x = ln |ax + b\ +2b

ax + b

b2ax + b 2 (ax + b)

1 2b( -3 )( + )"3 (n -2 ) (a x + b)"~2 (n - l)(ax + b)

ax + b dx 1 a 1- = ------+ In

x 1 (ax + b fdx I ax

a ^ = 7 b aTCt4dx x

b2 (ax + b) ab2x b3

Jx (ax + b) bx b

ax + b

ax + b

dx

J (x + a ) 2a (x + ) 2a

1 x+ a rc tg -

3x 3 x + ra rc tg

(x + a ) 4 a (x + a ) 8 a (x + a ) 8 adx 1 , x 1 , a - x 1 1 1 1 ;----- - = arcth = In-------, x < a

x ' - a a a 2a a + xdx 1 . x 1 . x a I 1 . I;----- - = arcth= In-------, x > a

x" - a a a 2a x + adx 2 l a x + b , ,

r----------- = . = a rc tg -p 4a c - b > 0ax +bx + c y l4 a c -b 2 %l4ac-b2

51

dx

3ax2 +bx + c ylb2 - 4ac \Ibz - 4 a c b2 - 4 a c

- 2 < 0 * dx 2 ,2 ;----------- = -------------, 4 - = 0

+ + 2 + * j 1 | I 2 | ------------ dx = In + ftx + c ------ I-3ax +bx + 2a ' ' 2 a 3 i

2 , 2ax + b-a rc th - ln

2a x + b - \ l b 2 - 4 ac

2 ax + 4 ac

3ax2 +bx + cr m x + n , m , I 2 , I 2an - bm 2ax + b , 2J - ;-------dx = ln |ax +bx + c |H----- , arctg / , 4 a c - b >0

I:

ax +bx + 2 a m x + n

a ^ 4 a c - b 2 !4 a c - b 2

ax +bx + c 2 a m x + n

i 2 a n - b m . 2 ax + b l2dx = ln |ax +bx + c \ ------ arcth , , 4a c - b < 0

3ax +bx + c 2 a r dx 2 ax + b

a\lb2 - 4 ac J b 2 - 4 ac

4ac - b

(2n - 3)2a

, . . . , | 2 , i 2a n - b m l2dx = ln |ax +bx + c | -----;-------- 77 , 4 a c - b = 0

a(2ax + b)

r + - f---------* L _} (ax2 +bx +* (ax2 +bx + )" (n - l)(4ac - b2)(ax2 +bx + )" ' (n - l)(4ac - b 2) 1 (ax2 +bx + )"

x , bx + 2c b(2n 3) j- dx

c - b 2) h a x 2 +bx + c)"-'- d x = -

! (ax2 +bx + c)n ( n - l ) ( 4 a c - b 2)(ax2 +bx + c)" ' (n - l)(4ac - b2) J (ax2dx

= In

I

x(ax +bx + c) 2c

dx

ax2 +bx +

b dx 2c 3ax2 +bx + c

= arcsin + C, Ixl < a x \ a J

f , = arcsin(x) + C, Ixl < 1

f . - = ln (x + Va2 + x 2 j + C, |x |< a J . ^ - = ln lx + Va2 - x 2 l + C, |x |> a Va2 + x 2 ' ' \ la2 - x 2 1

Js h c x d x = - c h c x j c h c x d x = - s h c x

(sh2 c x d x = s h 2 c x ~ fc h 2 c x d x = s h 2 c x + J 4c 2 J 4c 2

fsh cx 03 cn n 3

: fsh" cxdx = ----------sh"*1 cxchcx--------- I s h " 2 cxdx, n < 0, n -1J ( + 1) n + l J

fchcxdx = shcxch""cx + - - fch"-2cxdx, n > 03 cn n 3

: fch" cxdx = ------ - shcxch"fl - + ^ fch',+2 cxdx, n < 0, n -13 c(n + l) n + 1 J

r dx 1, , cx 1, c h c x -1 1, shcx--------= - l n t h = - I n --------------= - l n ---------------

J shcx 2 c shcx chcx + 1r dx 1 - = cthcx

J sh cx f dx 2 - = - arctg e

J chcx

= i l n

ch cx - 1

chcx + 1

52

1 , = -th c x ch

dx chcx n - 2 t dx------=-------------;---------- ------ ;---, (I# 1sh"cx c (n - l) sh " cx n - 1 'sh"

r d x s h e * n - 2 r d x----------= ------------------------ 1--------- , 1

J c h " cx c ( n - l ) c h " cx n - l J c h " ~ c x

rc h " c x , c h cx n - 1 rcW~2 cx ,----------d x = ------------------- -- + -------- I-------------- d x ,

J s h cx c ( n - m ) s h m cx n - m J s h m cx

( ch" cx , c h "+1 cx n - m + 2 t ch " cx , : ---------- d x = -----------------------:------ 1- -------------- ------ d x , 1

J s h " c x c t m - D s h 1 c x m - 1 J s h " cx

t ch"cx , ch" 1 cx n - 1 rch" 2cx ,: -------- dx = ------------------ ------ 1-------- ------- dx, m 1

J sh "cx c ( m - l)sh cx m - l J shm cx

rshmcx , sh'"-1 cx m - 1 rshm2cx-------- dx = ---------------- ------ h------- -----------dx,

J ch"oc c ( m -n )c h " cx m - n J ch" cx

rshmcx . sh"1*1 cx m - n + 2 e sh" c x . : -------- dx = --------------------- H------------ ----- dx, 1

J ch" cx c ( - l) c h " cx n - 1 J ch" cx

rsh'"cx . sh "1 cx m - 1 rshm2cx: -------- dx = ------------------ + ------- ----- , 1

J ch" cx c ( n - l)ch cx n - 1 J ch" cx

fx shcxdx = - x c h c x - -i-shcx J

\x ch cx dx = - x sh cx - -i- ch cxJ '

j th c x d x = - lnjchcxl

Icth cx dx = i In I sh cx Ij c i i

Ith2 cxdx = x - - th o c 1

fcth2 cx dx = x - - th cx J

fth" cxdx = ------ - th"_I cx + fth"~2 cxdx, 13 c ( n - l ) 3

fcth" cxdx = ------ - cth"-1 cx + fcth-2 cxdx, n 13 c ( n - 1) J

j shbx shcxdx = ^(bsbcxchbx - cchcxshbx), b2 2

[chbxchcxdx + ^ (bshbxchcx - cshcxchbx), b2 2 1 b -cjchb xsh cxd x = -(fcshfocshcx-cchbxchcx), b2 2

fsh(ax + b)sin(a: + d)dx = a ch(ax + f>)sin(cx + d) C sh(ax + b)cos(cx + d) 1 a +c a +c

fsh(ax + b)cos(cx + d)dx = U ch(njc + b)cos(cx + d)+ sh (as + b)sin(cx + d) J " -i- -- a +

53

fch(ax + b)sin(cx + d)dx = a -sh (ax + b)sin(cx + d) ch(flx + b)cos(cx + d)1 a +c a +c

fch(ox + b)cos(cx + d )d x = --sh(ax + b)cos(cx + d) + -C ch (ax + b)sin(oc + d)J a +c a +c

- > 0, .

f ( x ) [ , ]. = 0 < , < 2 0, [a , fa].

I = lim / ( ! ; , -)(, - ) = f f ( x )d x(,.-. . ) - * 0 *i - l a

I / ( ) [ , b]. > 0,

[ , , 6, max(xr - ^ , ) < 5,

i=i

f ( x ) , .

, (. 45).

j f ( x ) d x ,

, = = fa }( ) . .

, : .

F / [a; fa] j f ( x ) d x = F (b)-F(a).

a

/ , [ ; ].

f ( x ) [a; fa]. f ( x )

[a ;b ] , [ , ; 2] [ ; ] .

54

f i x ) () [ ; ], f { x )x .g (x ) .

f { x ) , V a :

J f ( x ) d x = j f ( x ) d x .

/ ( * ) g(x) [a; b],

f ( x ) < g ( x ) V x e [ a ; b \ . : j f ( x ) d x < jg (x)dx .

.1. f ( x ) [a; b ', < f ( x ) < [; b],

(m, = const). m { b - a ) < j f ( x ) d x < ( - ) .

j f ( x ) d x < |/( ) |< & , a / + B g(x)} ix = A \ f ( x ) d x + jg (x)dx .

f ( x ) [ ;* ] ,

j f ( x ) d x = j f ( x ) d x + j f ( x )d x .

f ( x ) = 1, j f ( x ) d x = jd x = b - a , >.

. f { x ) [a ; b ],

[ ; ], j f { x ) d x = f ( c ) ( b - a ) .

. / ( * ) , g(x) , g{x) ,

f f ( x ) g ( x ) d x = f ( ) jg ( x )d x , a

55

- = {) < u < |3, / ( ) < < ,

, ()

< < , J f ( x ) d x = |/[()] du*

56

/180*0,017453, /(1 8 0 60)0,000291, / (180 60 60) 0,000005 .

, ,

. 49

, . 49 .

,

. 49 a

: sin = /

: cos = /

sinP = fc/c, ,

: t g a = /

: c tg a = b/fl

,

: seca = c/b

: cosec a = / a

, 0 90 ( 0 /2 ).

. (. 50) . 0(0; 0).

( , ).

. : . .

57

. (

, ). , . , .

. 51 .

. 49 .

: , . : , .

, 1.

, .

, ( . ).

0(0 )

30(/6)

45( /4)

60( /3)

90(/2)

180()

270(/2)

sin 0 1/2 4 2 /2 4 1 /2 1 0 -1

cos 1 41/2 41 1/2 0 -1 0tg a 0 1 41 0 ctg 41 1 \/4 0 0sec 1 2(41 41 2 -1 cosec 2 4i 2 1 -1

sin cos tg a ctg

I 0 < < / 2 + + + +

II / 2 < < + - - -

III < 3 7 1 :/2 - - +

IV /2 < < 2 - + - -

. 50

58

I :

. 51

p sinp cosp tgp ctgP/2 + cos a -s in a -c tg a - t g a

+ -s in a -c o s a tga ctg a

/2 + -co s a sina -ctg a - tg a

2 + sina cosa tg a ctg a- a -s in a cosa - tg a -ctg a

/2- a cos a sina ctg a era P

- sina -c o s a - tg a -ctg a

/ 2- -c o s a -s in a ctg a tg a

2 - -s in a cosa - tg a -ctg a

sin2 + cos2 = 1 tg a x c tg a = l, /2, n e Z

l + tg2a = l/cos2a , a n/2 + n ,n e Z1 + ctg2a = l/sin2 , , n u Z

59

, .

sin I----- sin Vl - sin2

= V l-s in , tg a = . ctga = ----------------- V l-s in 2a sina

cosa , ,--------2 >/l -c o s 2 a cos ai a = V l-c o s a , tg a = ----------------- , ctga = -+V 1 -c o s 2 a

tga 1 tga 1ctga = ----- , sina = , cosa =* ) Jill IA , , C u e Ur " ^ lga -y/l + tg2a l + tg2a

ctg a tg a = -------, s in a = -------------ctg a 4/l + ctg2a

1 ctg a, cosa =

t-y/l + ctg^a

sin sin = 2 sin----- -cos-------

2 2_ . + -

cos + cos = 2 cos-------cos-------2 2

. + . - cosa -co sB = -2 s in ----- - s in ----- -

2 2tg a tg P = sin a ~P^ a ,p -t n/2 + , n e Z

cos a cos p sin (aP)

ctga + ctgP = ;------, , * , n e Z

(bcosx = yJa2 + b2

bsinx = \la2 + b2

x arcsin

Va2 +b J

sinasinp = - ( c o s ( a -p ) - c o s (a + P))2

cosacosp = -(c o s (a - p) + cos(a + P))2

sin ac0sp = - (s in (a + p) + s in (a -p ))

cosasinp = i (s in (a + p) - sin(a - P))2

60

sin(a + - ) + sin(p + - a ) + sin(a - + ) - sin(a + + )sinasinPsiny = -----------------------------------------------------------------------------------

4-co s (a + - ) + cos(P + - a ) + cos(a - + ) - cos(a + + )

.sin a sinp cos = ---------------------------------------------------------------------------------------4

sin (a + p - ) - sin (p + X - a ) + sin (a - p + ) - sin (a + p + ) sinacosPcosy = ---------------------------------------- -----------------------------------------

cos (a + p - y) + cos(p + - a ) + cos(a - p + y) + cos(a + P + y) cosacospcosy = -------------------------------------------------------------------------------------

(, . . )

sin2a = 2sin a cos asin3a = 3sina - 4sin3 asin 4 a = cosa^4sina - 8sin3 a )

sin 5a = 16sin5 a -2 0 sin3 a + 5sin a- i f

sin(Ha) = 2',_irT sin a-I-----i i l J

cos2a = cos2 a - sin2 a = 1 - 2sin2 a = 2cos! a -1 cos3a = 4cos3a - 3 c o s a cos 4 a = 8 cos4 a - 8 c o s 2a + l cos5a = 16coss a - 2 0 c o s 3a + 5cosa

2tg a tg2a = 2

1 - tg a 3 t g a - t g 3a

tg3a =l - 3 t g a4 t g a - 4 t g 3a

tg 4a = , , l - 6 t g a + tg a

tg4a - 1 0 t g 2a + 5tg5a = tg a 2 - --------------------

5tg a - lO tg a + 1

ctg2 a -1 ctg 2a = e

2 ctg a3 c tg a -c tg 3a

ctg 3a = 2------- f l -3 c tg a

ctg4 a - 6 c t g 2 a + 1ctg 4 a = S ------ - --------

4ctg a - 4 c t g a

, /2.

. II- c o s asm = , ----------- , 0 < < 2

2 V 2

61

/2

- < <

1 -c o s a sm a 1 -c o s a----------- = -------------= ------------ , 0 < < 1 + cosa 1 + cosa sina

tgI =

a ll + cosa sina 1 + cosactg = J ----------- = ------------ = ------------ , 0 < a <

2 V l - c o s a 1 co sa sina

2 t g | l - t g 2! 2 tg | l - t g 2|sina = ------- ; cosa = --------- ; tg a = ; ctga = ---------

2 oc 2 a 2 ot , ; al + t g - l + tg - l - t g y g ~2

l - c o s 2 a . , 3 s in a -s in 3 a

sin a = -2 4

3 -4 c o s 2 a + cos4a . . 1 0 sin a -5 s in 3 a + sin5asin a =

cos a = -

8 16l + cos2a , 3cosa + cos3a

2 4. 3 + 4cos2a + cos4a

cos a = ---------------------------8, 10cosa + 5cos3a + cos5a . , , l - c o s 4 a

cos a = ------------------------------------ sin acos a = -------------16 8

. , . 3 s in 2 a -s in 6 a . . . 3 -4 c o s 4 a + cos8asin acos a = --------------------- sin acos a = ---------------------------

32 128

sin(a + p) = sinacos(3cosasinp cos(a + P) = cos a cos P + sin asinp

tg (a P ) = ^ ~ ^ , , */ 2 + n + * / 2 + , - * / 2 + , n e Z IT tgatgp

ctg (a P ) = ct8 a c t6P + 1| > ^ + # , - ^ n , n e Z c tg P ctg a

.

shx = -isin(ix), chx = cos(ix), thx = -itg (ix); sh(ix) = isinx, ch(ix) = cosx, th(ix) = /tgx.

62

shx chx ch2 sh2x = l

ch(-Jt) = shx; sh (-x) = -s h x th (-x ) = - th x ; cth (-x) = -c th x

sh(x y) = sh xch chxsh ch (x y) = ch x ch sh x sh , , . thx th v

th(x v) = ------------ i-1 thxthy

cthxcthy + 1 c th (x v )= '

e th y l cthx

sh2x = 2 ch x sh x ^thx

1 - th 2 X

ch2x = ch2 x + sh2 x = 2ch2 x - l = l + 2sh2x = ' + 1 - th 2 x

, 2thxth2x = -------

1 + th x

cth 2x = (th x + cth x)2

c h 2 x - l sh2xthx = ----------- = ------------

sh2x l + ch2xch 2x sh 2x = (sh x + ch x)2

sh3x = 4sh3 x + 3shx ch3x = 4ch3x - 3 c h x, . 3 + th2x

th3x = th x --------- l + 3th x

sh 5x = 16sh! x + 20sh3 x + 5shx ch5x = 16ch5x - 2 0 c h 3x + 5chx

, th4x + 10th2x + 5th5x = th x --------------------------

5th x + lOth x + 1

ch(x + y ) - c h ( x - y )shxsh = ----------------------------

2sh(x + y) + s h (x -y )

sh x ch = ----------------------------2

ch(x + y) + c h ( x - y ) chxch y - ----------------------------

ch(x + y) - ch(x - y)th xth y = -----------------------------

ch(jc + y) + ch(x - y)

63

s h * s h y = 2 s h ^ c h ^

2 2

ch * + c h , = 2 c h ^ c h ^ 2 2

c h * - c h y = 2 s h ^ s h ^ 2 2

sh (x v)thjcth_y = -------------

ch xch y

, 2 ch2x + lch x = ------------

2, , c h 2 jc - l

sh x = -----------2

. j 2thxth x = -------

1 + th x:

sh * chx

shx + chx = e'

.

, , (, , ) .

, ., , . X

X. : X, X, X .

X, : . : X, X .

. , : {a, b, } , {, , , ... } .

, . 0 , 0. .

() , . U.

, .

1. , , . . , , : = .

64

2. , , : ) = !

( -1 ) - - - ( - + 1) (-1)---(- + 1) \ 1-2-3

= , 0 < < , = " = 1, +,' + ... + = 2

; + c ;+1 = *,', < <

, :

( + )" =" + '-' + ' 2 2 + ... + &" = - ' ' ,1=0

, , () - ,

65

*

( >):

:

(, + 2 +... + a j = ^ -------'? - , , + 2 +... + = ., !2!... !

, .

,. : -

( )2 = 2 2> + 2 a2 - b 2 =(a + b)(a-b)( + b - )2 = 2 + 2 + 2 + 2 - 2 - 2

( b)3 = 3 2 + 3ab2 3 a3 b 3 = ( b)(a2 +ab + b2)

( )4 = 4 + 43 + 622 43 + 4 4 - = ( - )( + )(2 +2)

-

" - " = ( - )("~' + "~2 + "'32 +... + d2br- 3 + ~2 + ") 2" - 2 =(a + b)(a2'-' - a2"~2b + a2"~3b2 - . . . - a 2fa2"3 + 2"-2 - 2), 2" - 2" = ( + )(2"-1 - 22 + a2"V - . . . - a2b2" '3 + ab2'"2 - 2- ' ), n e N2"*1 + 2"*' = ( + )(2" - 2~' + 222 - . . . - a2b2"2 + ab2"~ - 2" ), 2"+1 + 2+| = ( + b)(a2r- a 2 'b + a2,- 2b2 - . . . - a V " 2 -nab2'" 1 - 2"), n e N

1. ( - )2" = ( - )2" , N .2. ( - )2"*' = - (6 - )2"* ', N .

, .

0 11 1 12 1 2 13 1 3 3 14 1 4 4 15 1 5 10 10 5 16 1 6 15 20 15 6 17 1 7 21 35 35 21 7 18 1 8 28 56 70 56 28 8 19 1 9 36 84 126 126 84 36 9 110 1 10 45 120 210 252 210 120 45 10

(- , ,... , ; ) .

66

1. ( + ) 2".2. , , .3.

; 2.

( ) (. 52).: , , , , .

(, ) :

| |,| |,| |,| |,.

, , : = .

. ( )

(') . . ,

.

:

* =/ = |

, , , ,,..., (| .

, , () 2 (. 53).

>------------------ -' . . -

. 53 , , .

( ,; 2) = + ,? ,.

, , , : + = +,

+ { + ) = ( +) + , + 0 = ,

( ) = ( ), + = ( + ),

aa + ab = ( +), = - 0 = 0.

. 52

67

i , j , , (. 54);

\ \> zi ; 2 < 2 < z 2 b ;

a = x ti + y j + zjc , b = x 2i + y2j + z 2k,

a = U ,;> '1; 21),

: a +b = (x, + x2; y l + y2; z, + z ,), a - b = ( x , - x 2; y , - y2;z , - z 2),

aa = (a x ,; a y ,; a z ,),

l l= V x.2 + >'i2 + z i2- A (a,;a2;fl3) ,

B(b,; b2; b3) , = (fy - \2 - 2\ - ),

= /(, - , )2 + ( - 2 )2 + (3 - , )2. ,

(. 55). : a x f r = = 0 . -

. : || : a t t : 4--

, :

; ; . ( ) , ,

, ( ) = 0 .

a, b ,c ,d . :

, .

, , . (,,) = 0 (

). ( ), ,

, 2 , = Xfi + 2

, , , b = 0 = 0. , b, .

d : d = xta + 2 + 3. |,,2, 3} d .

68

2, 3 , , , , , 2 2 3 .

, . :

, , , , , . .

1. (. 56).

, . ,

, :

+ = .2. (. 57).

, . , .

+ = , ()

:

^| | +| | -2| o a |x |o b |xc o s

69

a = x j + y tj + z,fc, b = x 2i + y 2j + z2k,to a b = xtx2 + y , y2 + z,z2, a 2 = x\ + y* + z,2.

/>\ ,2 + , 2 +z,z2

s , = . cos\a,b = --------- -----V I \\\ \ / * , +, +Z,

() , .

() :

(, ) = 0 .

/.|||. * 0 , X > 0 (. 59), , X < 0 (. 60).

= ^ ( ) 2+ ( )2 = \ 1 < + 1 =1 ( ,; 2) X

(,;2), . . (,;2) = (1;2). X, :

( + ) = +| .

.:( + ) = + .

+ + 2

,

0 \ { ,

. 59

. . . .

, 0 / \

'

,

. 60

, ,

, (. 61).

b , [a b ]

a x b , |[abj| = |a||b|sin

70

( ), : [ , b] = Se .

[ , b] = - [b , ] ;

[( ), ] = [ , ( )] = [ , b j ;

[( + ),] = [ , ] + [& ,] ;

[[a,b],c] + [[b,c],a] + [[c,a],b] = 0 , R1 R7

[ , ] = 0 [, [, ]] = (, ) - (, )

|[,]| +(, )2 =||2|&|2 ([, ], ) = (, [6 , ])

, , ( , , ) ( ,,)

i, j, , = {,, yv z ,}, b = {x2,y 2,z 2}=> [, ] = i j

*i , z,

*2 2 Z2

, z, 2 Z2

vl 1 , z,

*, , X,

= 1 z. 2 z2

X , Z,

X , Z ,

*1 >1

*2

(, , ),,

& :(, 6, ) = (6 ) .

,

, , .(, , ) = V^b , ,, , (, , ) = ~Vab , , ,

, , , . ( , b , V = 0 . )

(. . , ), .

e = { w r zi}> = { 2 - 2 }- C = {x 3,y 3,z ,} , =>(,,) =\ z,

*2 z2 z3

, , .

, , .

: , , , : , .

, .

, .

, (, : , , . .).

: .

( ) .

, .

. .

, .

, , .

, .

, .

, (, , ).

72

, .

, , .

.

, ( , ; ).

, , , .

, .

, ' , ', , , , ', , , , , ', ', ' ( ).

: , ( ).

: , , .

, , , .

, () , , .

, : , , ( : , , ), , , , .

, . .

. .

, .

73

2)

: + + = (2 + 2 >0). :

1) By + = 0 ; + = OY;

3) + By = 0 ;4) = 0 ;5) = 0 OY.

: -7 + = , 7 - .

= (, ) .

(. 1): = + , b 0, , = tga, a ,( < a < ).

(. 2): + = 1,

, 0Y.

(. 2): x co sa + y s in a - p = 0 , a ,

; .

- + +

---------------- -

. 1

i-jA2 + :- = 0 .

1

sJ a 2 + 2

; , * 0 , = 0.

, ( ,, 1) (2, 2)

- _ 2 - 1

. *, * * 2 ,*

, (0; 0) ~ = ,

.

, (. 3, ).

(. 4, end) , ,

, . . 5 . _____________________

, , . . 5 . -----------------------------------L- . 3

74

, , , , , . . 5 > , .

, .

, .

, .

, .

,

. 6 : 3 6 ; 4 5. 1 8; 2 7. 3 5; 4 6 . 1 7; 2 8 . 1 5; 2 6 ; 3 7; 4 8 .

+ By + = 0 A .-B D = 0Dx + Ey + F = 0

= + = y = p x + q

+ By + = 0 A D + B E =0Dx + Ey + F = 0

= + = 1 = px + q

( ,, ) (2, 2)

d = ^j(xl - x 2f + ( y l - y 2)2

___________.

. 4

75

(0, 0)

+ By + = 0 | 0 +0 +\

U ' + b >

+ By + = 0 Dx + Ey + F = 0

I C - F /- 1 1

1 2 + 2

= + y = px + q

+ By + = 0 Dx + Ey + F = 0

1) = + y = px + q\

- tana = ------ .

1 + 2) + By + = 0 Dx + Ey + F = 0 :

\a d + b e \

J a 2 + b 2 J d 2 + e >

aIx + bIy + cl = 0 x ,2 ~ 21 a2C, ~ a,C2a2x + b2y + c2 = 0 ,2 ~ 2, " ~ 2,

= , + , .. b2 ~ b; .. ~ , = 2 + 2 >

1J*

r 1 N1

_P*T 1

, ,

, . ,

. , ,

. , ,

.

, . . 180.

76

180. .

. 7

b ( ), b .

( ), b .

b 180 ( ), b .

180 ( ), .

b ( ), b .

, , .

, , .

, .

, () , (. 8).

. , , , (. 9).

, , .

( ) , , (. 10).

, (. 10).

, ( ) (. 11).

, , , :

, (. 12); , , . .

. 8

, / \ ,

V\Ja ,

/ 1\,

. 9

. 10

77

[) , .

. , .

() .

, , (. 13). .

. .

. ,

, .

, . 180 (. 14).

90. ---------------- 90. . 14 90

180.

, . . 15 : 1 3; 2 3; 2 4; 4 1. ,

, .

. 15 : 1 2; 3 4.

. 13

= 180

. 15 180 (. 16).

(. 17). (. 18). : , ,

.

78

, () (. 19); , , .

, .

: ( . 19 ) ( . 19 ),

; ; :

+ >

+ >

+ >

, .

+ + = . (. 20):

= + ; , = + ; , = +

2 = + + , .

S = \ ah = \ bhb=\ ch (-21),

S = -absinY = -acsin$ = -bcsm a,2 2 2

S = ^ { - ) { - )( - ) ( ),

_ abcS = -^ - , 5 = , -. 19

(. 22).

: a2 =b2 +2 - 2cbcosa b2 = 2 + 2 -2accosP2 = b 2 +2 -2afccosy (. 23)

_ b : ------ = -------= -------= 2, R -

sin sinp sin (. 23).

. 21 . 22

79

, (. 24).

= ,,,, :

= ,,; = ,,; = ,, :

1. , .

2. , .

3. , .

,

, (. 25).

ZA = ZA,; ZB = Z B ,; ZC = ZC,

. 24

~ 111ZA = ZA, ZB = ZB, ZC = ZC,

80

1. , .

2. , , , .

3. , .

( < 90, . 26), ( = 90, . 27), (>90, . 28).

. 28

, (. 29), (. 30) (. 31).

. 29 . 30 . 31

81

, (. 32). .

, , ; ;

> , > ; ;

+ > ; = /2, + = /2 .

;2 +2 = 2, , b ;

. , , , + - =, =, \ = , = - , = - , = - , ,

b; hb mh R ,r , , ;

a = bsina = bcosy,

c = b siny = cosa,

fl = c t g a = c-ctgv,

c = a tg y = a c t g a

, ; ; ; ; .

, .

; (. 31)

; ~

= ------ ;2

, ; , ,

:

; = = 1 ' = ^ - .

( , ), - , , .

_ chc _ 2 siny2 " " 2

:

m = h = l = a 4 b / 2 ,

= !/'3, = -\//,

R = 2r, S = 2 -\//4.

82

,

.

,

2 : 1, :

O F = - A F3

1 = -

3

= - 3

. .

:

' 22 + 22 - 2

; , ,

:

a = ^ ^ 2(mj + ) - m'j, fna,mh,mi

, , , .

, :

mb = 22 + 2 2 1

,

(. 34).D

: , , (. 34):

. 35

EF\\AC, EF = -,2

, () 1/2 (. 35).

83

, , : SABC = 4SEBF.

,

, (. 36).

,

= ( ) (. 37).b

(. 37) .

. .

, .

, .

, ( ).

. , :

y]ab(a + b + ){ + b~c) _

84

: , , ( . 38) , ; ;

:

1 1 1 : \

, , , .

., 2 Shtl = , S , ,

. , :

h = -\j4a2 - 2, ;2a j3

- ;h = -

, , , . , :

2 ^ ( - ) ( - ) { - )

, , (. 39). .

.

6. 39

85

.

. abc _

: R = -----; R = ---------, , , -45 2 sin

, , , S .

, , (. 40).

().

:

;

+ ,------ , , , .

:

S 1 ( - )( - )( - )

\

, ,

, .

, , (. 41).

, , (. 42). , .

. 41

ABCD , :1. : AB = CD,AD = BC.2. : ZA = /, ZB = ZD.3.

: = , = OD.4. 180 :

ZA + ZB = 180, ZB + Z C = 180,Z C + ZD = 180, ZD + ZA = 180.

5. - ' : AB = CD, AB || CD. pUc. 42

86

6 . . : AC2 + BD2 = 2(2 +2).

: S = ah, S = abs ina, S = - AC BD sin/.2

ABCD , ZBAD + ZBCD = , ZABC + ZADC = .

, (. 43). , .

: :

[]||[], []||[} = = CD = AD;

[A C ]x [B D ]; : AC2 +BD2 = 42; .

: = 4

a2sina, S = AC BD.2

, (. 44). , .

: : = CD, AD = BC, [AB]||[CD],

[AD]||[BC]; 90: ZBAD = ZABC = ZBCD = ZADC = /2; :

a = 2 arctg(a/b), p = 2 arctg(b/a), a + p = 180. :

= BD; .

: = 2 ( + ).

: S = ab d = ylaz +b2 S = d2 sin(a/2)cos(a/2)

-------------------------- . : R = yj(a2 +b2)/2 ( ) , . ,

D , . 44 .

87

, (. 45). , .

: : = = CD = DA; 90: ZBAD = ZABC = ZBCD = ZADC = /2.

: d = -\ ( )

; ( )

.: P = 4a = 8r = 2^2 R .: S = a2 = d lj2. : S = 2 = 42 = 2R2, R :

; = /2 .

: ; R = dj2 = (V2/2).

- .

, (. 46).

. ( ).

. : [ a d ]||[BC], :

(. 47), , , ( );

. :

(. 46): [iiFjIlljAD], EF = (a + b)/2.

: S = (a + b)/i/2, S = EF-h. ,

, , (. 48, ).

, / , / , (. 4 8 ,6). /

- 1, (. 48, ). . 47

88

, (. 48, ).

. 48

. 49

/ ' \/ / V j \/ V \

, , (. 49). . , . . , . (. 50): = /2 = //2 .

. 50

, , (. 51).

, (. 52 ).

, : = 2" 1 , ... , ,, 2,..., pt

= 2 +1 (s ). = 3, 4, 5, 6 , 8 , 10, 12, 15 16, 17, 20, 24, 32, 34, ...

= 7 ,9 ,1 1 ,1 3 , 14, 18,19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 3 3 ,... : 3, 5,17, 257,65537. ,

(. 53).

89

, (. 54).

, .

, . . .

, 180. . .

, .

. 0 , a R , 0 , - :

( 2 . ( 2 .; = 0 +.KCOS

90

, 2 , 1 2,S, S2 ,

, : ,= , S, :S , = 2.

. 56

, , , , (. 56).

, , .

, .

, ( . 56 ).

, , ( . 56 ).

, R

:

= 2Ksin.2

. 58

, (. 57). , , , .

: L R ( ) L = ;

; L = 2nR------ .

F 3601 ,

, .

. ,

. .

91

: ,

; , ,

, (. 58); , ,

, . ,

( . 58 '; . 59 ). ( . 60 ZAOB ).

, . , ,

(. 61, /. ). , .

: , , ; , , ;

;

, . , ,

(. 62). , ,

(. 62). , .

, .

, : /A B C = ^ / , ZADC = n - ^ Z A O C (. 63, ).

: = KD, = 2rsin(a/2) (. 63, ).

: AC -AD = AF = 2 (. 64).

: = 2 = d. : S = 2 = nd2 / 4 . ( ): / = . ( ): SK = a r 2/2.

( (5 ): / = /180. ( ): SeK = 2/360. , :

D 6. 63 . 64

92

, . , .

.

. .

, , , .

, , , , .

, .

, , .

, : , , ; , ,

.

, .

-

= r0 + at, 0 ( 0) = 0 (0; 0; z0) , ; = (1; ; ) . ( ): x = x0 +lt, y = y0 +mt, z = z6 +nt.

~ ~ Z~ Z0 I m

x ~ xa ~ z ~ zox , ~ xo 1 - 0 z, ~ zo

fAlx + Bly + C,z + Dl =0,s , , - [A2x + B2y + C2z + D2 = 0 * ' *

= = .2 2 2

- ---- 1 ( ^1 1 \ 1 = . , = ; ;L 1 [ 2 2 2 2 2 2 )

, = (,; ,; ,); 2 = (2; 2;2)

93

, .

, ,

, . .

, .

, , . , , . , ,

, .

.

.

()

. 65

Ax + By + Cz + D = , , , , D , , , . (. 65):(, JV) + D = , - M(x\y\z), N = (; ; ) (). N

' J a 2

COSp =

+-

+ 2

4 2 + 2

+ 2

/7cosy =

' ' + 2 + 2 , . D = . = 0 ( = , = 0 ) ( OY, OZ). = = ( = = 0, = = 0 ) OXY ( QXZ, OYZ).Z = 0 ; = 0 Oxz; = 0 Oyz.

94

+ + = 1, a = D/A\ b = - D / B ; c = D/C,

, OX, OY, OZ.

, M(xB;y 0;z0) N(A; ; )

A ( x - x 0) + B ( y - y 0) + C(.z-za); :(( ),0 = 0 .

, (: ;

( ( - , ) ,^ - ,), (, - ,) (),

- , - z " z, * 2 - * , 2 ~, Z2 -Z . * - * > - , Z3 ~ Z,

= 0 (

= 0 .

()

xco sa + 7 cosP + z c o s y -p = 0 ; :( , ) - = , , . : + By + Cz + D---- < =-\2 + 2 + 2

. ' = - 1 2 + 2 + 2

, D, D * 0, , D = 0.

, = r\ + a,f = 2 + a2t :

( - ,) , 2 = 0 ,1

( r - r 2)a,a2 = 0 .; , = ({,, m,, ,), - , - , z - z ,

/, , ,

f2 12 2

2 =(/2,m 2,n2), ; = (x 1,y , ,z 1),

= 0 .

, = ^ + at = r2 + a t :

( - ,) ( 2 - ; ) = 0 . x = xt +lt x = x2 +lt, z = z1+nt z = z2 +nt, - , - , Z - Z ,

* 2 - * l Z2 Z, /

: = , + mf = :

= 0 .

95

,

, .

, , . , , . , , , .

, .

, . , . , , . , , .

, ( , , , , ) 90. ,

, . ,

, , .

, .

d = *0 cosa + C0SP + zo cosY - p|>i \^xa + By0 +Cz0 +D\

-J a 2 + b 2 + c 2

5 = x0 cosa + y0 cosP + z0 cosy - p, P Ax0 +By+Cz0 +D

! 2 + 2 + 2 D, D * 0 , , D = 0.

96

( )

, 7 + D, = 0, 2 7 + D2 = 0, , = 2 (, ||2), D , * \ D 2. Alx + B1y + Clz + Dl = 0 A2x + B2y + C2z + D2 =0, 1 _ , _ , Dj, 2 2 D2 '

:

, , .

, .

, , .

, , .

.

,

Ax + By + Cz + Dt = 0 Ax + By + Cz + D2 = 0 :

, t e - A l

/2 + 2 + 2 ,

\ [ - (-||) = 0 ( - , ) = 0 : d = -------j:-------.

, 7 + Dl = 0, 2 7 + D2 = 0,

.2 AjX + Bty + Ctz + D, = 0 2 + 2 + C2z + D2 = 0,

-i = = ] ( - i ).

, 7 + D, = 0, 7 + D2 = 0, , = \2, D, = XD2. Alx + Bly + CIz + DI = 0 A2x + B2y + C2z + D2 =0, = bl = c l = d l

^2 ^2 ^2 ^2

( )

12 + {2 + ,2 = 0, (Np N2) = 0.

97

Ax + By + Cz + D, = 0 + By+ Cz + D2 = 0,

cos z 2 z i , ,

d =z2 - , 2 - ,

, /,

* 2 " * . - , /, ,

(V ',2 + ,2 +2)

? = ;+, = ^ + a ,t , , II2 11

, I, , ,

h 2 2 2 1 2 \ 2 2 Z 1

7 = rx + t = 2 +a1t, , 2 - (f2 -rI)ata2 =0.

,

h h 2

2 ~ 1 2~, Z 2 Z , , , ,/, , ,

= 0.

98

r = 7 t +a}t =r2 +a2t, , . (^ -^ ) , 2 #.

,

2 - 1 Z 2 " Z 1 m, , ^ 0 . d =

mod

: d =

7

- *

h

2

2 - \ 1 .

[ ,2]

z2 -z ,

,,

1 1i h+ +

m2 n2 i2

( )

cos(p/-^\ ar a2 /1/2+w,w2 +

= cos [, , 2 = rr jrr - = , )

, 0. , / + + 0.

, - 0,-70 +D = 0 . , 1 + + 0, * 0 + ByQ + Cz0 + D = 0.

, - = 0,-70 + 0. , / + + 0, 0 + 0 + Cz0 + D 0. :- - A B C \\ = = .

/

sinq>= cos1\Al + Bm + Cn\

_ _ - n r 0 +D r = r 0 - a .

:

x = x0 +ltr y = ya +mtv z = z0 +nf,, f , = -Ax0 + By0 +Cz0 +D

Al + Bm + Cn

99

, 0(0) - - n r + D -

=7Q+nt. : = 0 +At, = + 1. z = z0 + Ct.

, , , , (. 66). , .

, , , (. 66).

, , , .

: arccos(l/3) (70,53); /2 (90 ); Jt-arccos(l/3) (109,47); 2 -arctg () (116,56);

2 arctg(cp + l) (138,19), /5)/2 .

, , (. 67).

, , ,, , ,

, .

. ,

. .

. 67

.

100

: cosa = cosPcosy + sin(3sinycos cos = - cos cos + sin sin cos a , , , , , , , .

_ sina sinp sin ;-------= ------- = ------- ,

sin A sin sin , , ; , , .

, , ( ) , (. 68 ).

.

.

,

5 , : 1 = .

, , 2.

, 4 - . 68 ( ).

, , ,

, (. 69).

, , ( . 69 ABCDE, KLMNP).

, . ( . 69 ABLK, BCML, CDNM, DEPN, ).

.

.

101

( . 69 , BL, , DN, ).

, ( . 69 - KR).

, , ( . 69 EL).

, .

. , , , ( . 69 EBLP).

, .

: . . . .

. .

Sp = 2S,*,, + S6c,K, Sx ;S6oK ; S6oK=pl; ; I .

V = QH, V = Q,/, Q ; , Q, .

, , .

:

( ). .

:

.

. ,

, . , (

), . :

. . .

, (. 70). , , .

102

, , .

, , .

, , ( . 70 DBi ).

, , .

: .

, , , ; , :

, BD, = , = DB, =d,

d2 = 2 +b2 + 2. .

.

, .

: S6 = 2( + b)t , b ,

. 5 = 2{ab + + ).: V = abc, , , . ,

. : S6=P0x.h, 0 , h . : S = S6 -I- 2S0, S0 . V=S0xh. .

. : S6 = 42, . 5 = 62. V = 3.

, , , (. 71).

, . .

, (. 71 ).

103

, . . ,

. ,

( ).

, .

, .

: 5 . 71 :

, ;

; ,

, , . :

, ;

;

.

V = SAi, S h .

( ): Sb = .-

( ):

. , ( ). ,

.

. . ,

(, ) ( ).

. . ,

, . ,

( ).

104

. , , . ,

( ).

. . ,

, , , . ,

( ). . ,

, . ,

( ).

, , .

: ;

; ,

; ,

, /, ;

:

Sb = i p a = ^b2 sin , , ,

, b , .

, . .

, , .

:

V = ~(Qi + /Q, Q2 + Q2 J, h ; Q,, Q2 .

: Sc = ^(, + 2) /i6oK, ,, 2

; h6oK . .

.

105

,

, (. 72).

, () , ().

. S{riH = 2nRH. Su = 2nRH + 2kR2. V = R2H. . 72

, . (. 72).

, 2/ 2 + 1 j b 2 =1 (. 73).

, 2/2 - 2 / b 2 =1 (. 74) :

2 v2= ( > 0 , > 0 )

-

, 1 = 2 , > 0 ( )(. 75).

.

, , ( ) (. 76).

: , , ().

: , .

, .

, .

106

() .

.

.

.

271^ 1-cosy j, (

). S6oK = nRL.

SK0K = nRL + nR2. S. >= = nRL + nR2.

V = - kR2H.3

, ( ) . , , .

() , .

, . ( )

, ( ).

, . , (

). , (

). , ,

, (. 77).

. 76

107

S6oK = ( + )/. SIm;IH = nR2 + 2 + n(R + r)l.

V = nH(R2 +Rr + r2).3

( , , ) (. 78).

. 78: : , 6 , .

, , (. 79).

. :

. , .

. ,

. .

5 = 4nR2.

V = -n R \3

S = 2nRH ( ).

V = 3 ( - -

V = nR2H.

. 79

, . X X ' Y' , X Y = X'Y'.

, , . . , ,

, , .: ;

, ; - ; .

108

. 81

F'

, ', ' (. 80).

Z A.

.

, (. 81).

, .

F F', ', , . F F' (. 82).

. 82

'

. 83

'\

. 84

( ), , (. 83).

I , A e l . ' /, ' = ' X 1 (. 84).

I F F', ', /.

F F' / (. 84).

I , /, / (. 85).

. 85

. , ,

109

. 86

, (. 86 ).

, , .

, (. 87).

, , a W , ' , .

(; ) F '( + ; + b) F'. F, (; ) ( + ; + ), (; ), ' = (, ).

. 87

, , ', |''| = :||, , (. 88 ).

( ) = 1.

F F' , . - / 1 (. 89). F

__!____ F '' F', '' = .

. ,

, ', ', ' , ' ' '.

, , , .

, , , , .

.

. 89

110

* 1, , - .

D .

, (. ).

(, ). , ( ).

Ft F2, F2 F3, F, F3 (. 89).

.

( * 1) ( ), X X ', ' = (. 90).

.

. .

, 1, : . , 0, : . ( , - 1) .

, , .

()

, L , metre (, )

,

, , kilogram (, kg)

. , , : , ; ,

, , second (, s)

-

, I , ampere ()

, , 1 2

, 0 ,

,kelvin()

, , .

, / , candela (, cd)

, , , 1

, N

, mole (, mol)

, , . , (, , , )

112

(

)

(, rad)

, ( ), ( )

(, sr)

, , ( ) ,

(

)

, / . V ,

, hertz(, Hz)

, , ,

, F , newton (, N)

,

, , W , joule (, J)

,

, N , watt (, W)

, , ,

, , pascal (, )

, F , S

, O v

, lumen(, 1)

,

,

, lux (, 1)

, ,

, q, Q

, coulomb (, )

, ; ,

(), 17, .

, volt(, V)

,

113

, R

, ohm(, )

, , ,

,

, farad (> F)

,

,v /)

, weber(6 , Wb)

, -

, tesla (, )

, ,

, henry (, )

, , , ,

, siemens (, S)

,

( )

, becquerel (, Bq)

,

, gray (, Gy)

, ,

, sievert(, Sv)

, .

, katal (, kat)

,

114

,

() 1,85318

() 1,852

() 1,60934

() 185,2

914,4

304,8

25,4

(1/10 ) 2,54

(1/100 ) 254

() 2,58999 2

4046,86 2 = 0,404686

0,836127 2

929,030 2

1233,48 3

2,83168 3

0,764555 3

28,3169

16,3871 3

() 158,987 3

() 115,627 3

() 36,3687 3

() 35,2391 3

() 4,54609 3

() 4,40488 3

() 3,78541 3

() 1,1361 3

() 1,10122 3

() 0,946353 3

() 28,413 3

115

/

10' deca da

102 hecto h

103 kilo k

106 Mega

9 Giga G

12 Tera

15 Peta

10 Exa

21 Zetta 3 Z -

24 Yotta Y

-

10"' deci d

-2 centi

10~3 milli

micro ,

10'9 nano

12 pico

10"15 femto f

-'8 atto

-21 zepto 3 Z

-2" yocto

1 = 10241 = 210 = 1024 1 = 10242 = 2 = 1048 576 1 = 10243 = 230 = 1073741824 1 = 1024'' = 240 = 1099 511627 776 1 = 10245 = 250 = 1 125899906842624 1 = 10246 = 2 = 115292] 504606846976 1 = 10247 = 270 = 1 180591620717411303424 1 = 1024 = 20 = 1208925819614629 174706 176

116

, , .

( ) : , , , , , , , , , , , .

: , , . , ,

. ,

. .

, . .

, , .

q l,... qk. , ,

, . : ,

, , , , , , .

: .

.

: , .

: ( ); ( ); (- -).

, . . .

, , ().

, .

, . , : ; ( ).

, .

, .

117

, . *

, () .

, .

, . , . ,

.

( = 0, v = const): s = v t , , v (/), ( ().

a t 2 ( = const): s = v 0t + , s = ------- ,

2 2 0 (/), (/2), t ().

, .

( = 0, v = const): v = A s / A t , As (), At ().

( = const): v = v(1+at, v = +2as, v0 (/), J (), (/2). t ().

, As ds

v = J'm0 =

(), At (). ,

(): v = , I (), At At

().: -

s, (): = ~ >

As (), At (). , .

.

; () ( = 0), - - ( 2 ( = ): to = , = = 2nv, -

t - (), At (), (), v (-1).

( = const): Et2

) = 0+ ( ( - t 0),

118

-.. ( ).

, -(

= > - At

/; At (). ,

(), . (. 1).

, ^ ,

( ) ( ): F = aum.

, . -

V 2 >2 : = , = , 4

(/), (1/), - ().

119

. (. 2),

() ^ ). = + = * A t

(/), At (). = const

: - ^ - - (1/), - At At

1/2. -

1 : 1 = ,v

v "1. ,

, v = ,

. ,

. . 2

(, ): 7 = ^ ( w (r (2), . i- , . i- .

: / = j r 2dm = j p - r 2dV, dm = p d V dV , , d V .

, / = j r 2dV.

( ) , .

; : - mv, (), v (/).

( , , , ) . , , .

, .

L : L = r x p = r x , - , , (), v (/).

: Ll

, : L = I(bt I , .

, L (. . ).

120

: = 1 + ----- , I

L : L = |r| | | sin 0^, 9 , , , .

(L) ():

= = F, F (). dt

, , . , , , . F = , (), (/2).

( ) () -, , :

_ _ _ - d lM = r x F , = FI = F rs in a , M = , F (), -

(), I , 1 , (), a F .

.dl

dt , (1/), .

, (. 3).

FK = -2 ( ) - = 2 ( ), , , v .

: = 2[, ].

, (. 4).

, F .

, F .

, , . , ( ).

, :

Fml , - :

. 4

. 3

121

Fa6 + F ( ),

: Fu6 = , FK = 2 [ , ], ' .

, .

, , , . , . , , . .

, -

, : = -, , ; /

; ; 2.

= !]= = ^ - = / , L - dt dt

; .

, . . .

. 5: R , ,N .

, F, S '

F : = , F (), S (2).

S . 5 (): = m ( g a ) ,

(), g (/2), (/2).

( 2"! () : P = m \ g , m

(), (/2), v (/), ().

() .

: ,

, ;

, / ;

, . , . , .

122

: = , (