ﻢﻴﺣﺮﻟﺍ ﻦﻤﺣﺮﻟﺍ ﷲﺍ ﻢﺴﺑ -...
TRANSCRIPT
بسم االله الرحمن الرحيم والصلاة والسلام على أشرف المخلوقين محمد سيد المرسلين وعلى آله وصحبه أجمعين
ملخصات مع تقنيات أما بعد ٬ يسرني أن أقدم لكم هذا العمل المتواضع وهو عبارة على
مجمعة في كتاب واحد بكالوريا علوم تجريبية ثانية الرياضيات لمستوى ال
وهي للأستاذ حميد بوعيون
sefroumaths.site.voila.fr
تجميع وترتيب
ALMOHANNAD
لا یت وا
I ( 1 ( :
2 ( :
( )
a
a
a
+∞ + = +∞−∞ + = −∞+∞ + ∞ = +∞ ∈−∞ − ∞ = −∞+∞ − ∞
( )0
0
a a× ∞ = ∞ ≠∞ × ∞ = ∞
× ∞
00
00
0
a a
a
∞ ≠= ∞ = = ∞∞
∞∞
3 ( !:
(a ( )( )lim
f x
g x∞
∞=∞
.
(b ( ) ( )( )lim f x g x∞
+ = +∞ − ∞
(* ( )f x ( )g x .
(* ( )f x ( )g x .
(c ( ) ( )( )0
00 lim
0 0x
f x a aa
g x
−≠ = = .
(d ( ) ( )( )0
0 0 00 lim
0 0x
f xa
g x
+≠ = = ! " .
e("# :2x x= $
2
2
; 0
; 0
x x x
x x x
= ≥
= − ≤
4 ( . ( )
( )( ) ( )
20 0 0
1 cos tan sin1lim lim 1 lim 1
2x x x
ax ax ax
ax axax→ → →
−= = =
5 (%. (a% & f ' 0x() & ( )
0
limx
f x &
%( ) ( )0
0limx
f x f x=* f ' 0x.
(b+& f ,' -. /0 1 2 3& 3" . /0 ' 3&* -.4 ' (
( .
6 (%! f 0x 5 % & f
'20x& () ( )0
limx
f x& ( )0
limx
f x l= ∈
*f g '2 0x 6
:( ) ( )( )
0
0
,g x f x x x
g x l
= ≠ =
7 ( & . (a ( ) ( )f x l g x− ≤ 0x
( )0
lim 0x
g x =
(b ( ) ( )f x g x≤ 0x ( )
0
limx
f x = +∞
(c ( ) ( )f x g x≤ 0x ( )
0
limx
g x = −∞
(d+& ( ) ( ) ( )g x f x h x≤ ≤ 0x ( ) ( )
0 0
lim limx x
g x h x l= =
II (% ! ' &%. 1 ((a 7 ' 8' .
(b' 9: 8' 9: 7 . 2 ((a+& f* . ' :
(*[ ]( ) ( ) ( ), ,f a b f a f b= (* ] ]( ) ( ), lim ,a
f a b f f b+
= .
(b +& f* ':& ' :
(*[ ]( ) ( ) ( ), ,f a b f b f a= (* ] [( ), lim , limb a
f a b f f− +
= .
3 ((& ) *! (a & +f/0 ' [ ],a b
λ ' 0 ( )f a ( )f b
(b & +f/0 ' [ ],a b
( ) ( ) 0f a f b⋅ ⟨ &( ) 0f x = ; ] [,a b
"#: (* ( ) ( ) 0f a f b⋅ ≤ * [ ],c a b∈. (*+& f 9: * c .
III ( 1( +& : (*f /0 ' I
(*f /0 9: I (*( )f I J=
f ) 0 1 :f J I− → & :
( )( ) ( ) ( )1:x J y I f x y f y x−∀ ∈ ∀ ∈ = ⇔ = 2 ((a 1f −/0 ' J
(b 1f −/0 9: J <"& 3 f. (c ..&& fC 1f
C − 6'& )& ;
( ) : y x∆ =.
*( )0
limx
f x l=
* ( )0
limx
g x = +∞
* ( )0
limx
f x = −∞
* ( )0
limx
f x l=
*[ ]( ) ( ), :c a b f c λ∃ ∈ =
*] [( ) ( ), : 0c a b f c∃ ∈ =
*f I& J
0
0
∞∞
0∞× +∞ − ∞
3 ( +1f −
. +& f /0 9: =; : I
( ) ( ): 0x I f x′∀ ∈ ≠* 1f −/0 =; : ( )J f I=
( ) ( ) ( ) ( )( )1
1
1:x J f x
f f x−
−′∀ ∈ =
′
IV (! , ' n ( )*n∈ 1 (-: x +
n x 7 y IR +
1 ny x=.
(* :4 16 2= 42 0 2 ≤و16 =.
(* ( )42 16− = 4 16 2≠ − 2 +− ∉
2 (%. (a n/0 +
(b ( ) : 0nx x+∀ ∈ ≥
c( ( ), : )*
)*
n n
n n
x y x y x y
x y x y
+∀ ∈ = ⇔ =
< ⇔ <
.
(d ( ), : )*
)*
n n
n n
x y x y x y
x y x y
+∀ ∈ = ⇔ =
⟨ ⇔ ⟨
.
(e n* 1 : ( ), : )*
)*
n n
n n
x y x y x y
x y x y
∀ ∈ = ⇔ =
⟨ ⇔ ⟨
(f n* . :( ), : )*
)*
n n
n n
x y x y x y
x y x y
∀ ∈ = ⇔ =
⟨ ⇔ ⟨
(g (* ( ) ( )0 .n
nn nx x x x∀ ≥ = =
(* n . ( ): n nx IR x x∀ ∈ =
(h n p *IN a b +
(*.n n na b ab=
(*( );pnp np pn na a a a= =
(*( ); 0n
p npn nn
a aa a b
bb= ⟩ =
(*.npp n pn a a a +=
(i (*( ) ( )* , 0 :p
pnnn p x x x∈ ∈ ∀ ⟩ =
(* p. :( ) :p
pn nx x x∀ ∈ =. "#:
1 ( 0xy ⟩* n n nx y x y= ⋅ n
nn
xx
y y=
(2 (* ( ) 330 :x x x∀ ≥ = ( )
( )
333 3
333 3
; 0
; 0
x x x x
x x x x
= = ≥ = − − = − − ≤
.
(* 3 3 3 3
2 2 2 2
a b a ba b a b
a ab b a ab b
+ −+ = − =− + + +
4 4
3 2 2 3
a ba b
a a b ab b
−− =+ + +
(j a b *IR+ r 'r
r r r ra a a′ ′+⋅ = ( )rr r ra a
′ ′=
r
r rr
aa
a′−
′ = ( )r r rab a b= ⋅
1 r
ra
a−=
r r
r
a a
b b =
ت ادیا
(I . 1 ( :
U I :
( ):U I
n u n
→→
2 ( : :
( )n n IU
∈:
(a M ( ) nn I U M∀ ∈ ≤.
(b "# m ( ) nn I U m∀ ∈ ≥.
(c$ "# % . mM ( ) : nn I m u M∀ ∈ ≤ ≤.
: ( )n n I
U∈
0k ≥ ( ) : nn I U k∀ ∈ ≤
3 ( : :
( )n nU
∈:
(a ( ) 1n nn U U +∀ ∈ ≤.
(b $& ( ) 1n nn U U +∀ ∈ ⟨.
(c #& ( ) 1n nn U U +∀ ∈ ≥.
(d $& #& ( ) 1n nn U U +∀ ∈ ⟩.
(e ' ( ) 1n nn U U +∀ ∈ =.
: (1 % ( )nU( p np n u u⟨ ⇒ ≤.
(2 % ( )nU( #& p np n u u⟨ ⇒ ≥.
(3 ) ( )nU * +
1n nu u+ −. (*% 1 0n nu u+ − ≥( ( )nU .
(*% 1 0n nu u+ − ⟩( ( )nU $& .
(*% 1 0n nu u+ − ≤( ( )nU#& .
(*% 1 0n nu u+ − ⟨( ( )nU $& #& .
(*% 1 0n nu u+ − =( ( )nU ' .
(II 1( : ( )n n
U∈
r
( ) 1: n nn U U r+∀ ∈ = + r . ) / . :
(a ( )nU % ' $ .. 0 1 ' 21 .
(b) ( )nU3 + 1n nu u+ − 1n nu u cte+ − =. 0 1 ' .
(cو 0 $ و 4' 3 1
2a c b+ = $ 2
a bb
+ =.
2 ( . ( )nu 5 ) r0 1 0u
0nU U nr= + ( )n∀ ∈
: 1 (1 0 1u1 + $ ( :
( )1 1nU U n r= + −.
2 (1 0 2u1 + $ ( : ( )2 2nU U n r= + −.
3 ( 6# : p nU Uو
5 )r( ( )n pU U n p r= + − ) 3pوn+5 8 .(
3 ( : ( )nU 5 ) r 0 1 0u :
( ) ( )00 1 2 ... 1
2n
n
u uS u u u u n
+= + + + + = +
0u 0 9: S
nu ;0 9: S 1n + 9 S .
:
1 (1
1 2 ....2
nn
u uu u u n
++ + + =.
2 (0 10 1 1....
2n
n
u uu u u n −
−+
+ + + =.
3( 6#
( )1 .... 12
p nP p n
u uu u u n p+
++ + + = − +.
(III . (1 :
( )n nu
∈ 1 q
: ( ) 1: .n nn U qU+∀ ∈ = $q . ) /
: (a )$ 8 1 ( 1
. 0 1 ' 21 % ' $ < ;. (b ) ( )nU3 1 1nU +
=nU 1n nu q u+ = ⋅. (c0 cوbوa $ 4' 3 1
1 2ac b=.
2 ( : ( )nu 5 ) 1 q0 1 0u
( ) 0: nnn u u q∀ ∈ = ⋅.
: 1 (1 0 1u1 + $ ( 1
n pnu u q −= ⋅
2 ( 6#: n pu 1 uو
5 )q ( n p
n pu u q −= ⋅ )3pوn +5 8 .(
3 ( : ( )nU 5 ) 1 q0 1 0U.
>( 1)q ≠.
1
0 1 0
1.... .
1
n
n
qS U U U U
q
+−= + + + =−
0u :0 9: S .
( )1n + : 9 S. :
1 ( 1q = ( ( )0 1 0.... 1nS u u u n u= + + + = +
2 (0 1 1 0
1.... : 1
1
n
n
qu u u u q
q−−+ + + = ⋅ ≠−.
1 2 1
1....
1
n
n
qu u u u
q
−+ + + = ⋅−
6#
1
1
1....
1
n p
p p n p
qu u u u
q
− +
+−+ + + =
−
(IV . 1 (lim nq
0 ; 1 1
1 ; 1lim
; 1
. 1
n
q
q
دة q
− ⟨ ⟨ == +∞ ⟩ ≤ −
2 ( !". (a ( )nU ( )nV n nU l V− ≤ ?# &4 .
lim 0 limn nV U l= ⇒ =
(b ( )nU ( )nV n nU V≤ ?# &4 lim limn nU V= +∞ ⇒ = +∞ lim limn nV U= −∞ ⇒ = −∞
(c ( )nU ( )nV ( )nW n n nV U W≤ ≤ &4 ?#.
lim lim limn n nV W l U l= = ⇒ =
3 ( ( )nU 5 5 % . @;0 %= 5 .
4 ( .
(a ( )nU ( )nV n nU V≤ ) )n nU V⟨ (
?# &4% ( )nU ( )nV
(lim limn nU V≤. (b . (c "# #& .
5 ( ( )1n nU f U+ =
f/: $ I $ ( )0
1n n
U I
U f U+
∈ =
(*% ( )f I I⊂$ ( .
(*% f/: :# I ( )nU 5 5 ( l
( )f l l=
I( . 1 ((a ( ) 2 2/ , 1z a ib a b =iو = + ∈ = −
(b z z a ib= + aوb ∈
i ( ) 2 1i i∉ = −
(c (* z a ib= + ! " ! # z. (*a % &'! # z ( )oR z a′ =
(*b % ( &'! # z ( )Im z b= (* )*0b =+, z a= ∈. (* )*0a =+, z ib= ∈* z- % ( . 2 ( a b a′ b′ . α ∈
a a
a ib a ibb b
′=′ ′+ = + ⇔ ′=
00
0
aa ib
b
=+ = ⇔ =
II ( . 1 ( - z a ib= + . aوb ∈
, %#z / ' ) z% - z a ib= −. 2 (0 (.
(a z z z
z z z i
= ⇔ ∈= − ⇔ ∈
(b z z z z
z z
′ ′= ⇔ ==
(c 1 2 1 2.... ....n n
z z z z
z z z z z z
′ ′⋅ = +
+ + + = + + + (d
( )1 2 1 2.... ....n n
n n
z z z z
z z z z z z
z z n
′ ′⋅ = ⋅
= ⋅ ⋅
= ∈
(e 1 1
z z =
z z
z z = ′ ′
(f z x iy= + ( )
( )2 2
2 2 Im
z z x Ré z
z z iy i z
+ = =
− = = 2 2zz x y= +
III ( 1 ( : z a ib= + . aوb ∈
%#z / ' ) ! % z -
% : 2 2z zz a b= = +
2 (":
(a )* z a= ∈ +, z a=
)* z ib= ∈ +, z b=
(b 2 2 2z zz a b= = + z z z= = −
(c 0 0z z= ⇔ = z z z z′ ′+ ≤ +
(d 1 2 1 2..... ....n n
zz z z
z z z z z z
′ ′=
=
nnz z=
(e 1 1
z z=
zz
z z=
′ ′
#$% :. # ! % :
(* 2'
z zz zz
z z z z
′ ′= =
′ ′ ′
(*( )( )( )( )
( )( )2 2
a ib c id a ib c ida ib
c id c id c id c d
+ − + −+ = =+ + − +
IV ( &'( ) . 2# " 34P56 5 * # ( )1 2, ,o e e
1 ( :
(a ( ),M x y P z a ib= + # M
( )aff M.
(b ( ),u x y
v
z a ib= +7! # u
( )aff u z=
(c z a ib= + ( ),M x y # z
%,P ( )M z.
(d z a ib= + 7! ( ),u x y
# z
%,2v ( )u z.
#$%: (a ( ) ( ) ( )2 1. 1 . 0aff e i aff e aff o= = =
(b
( ) ( )( ) [ )( ) ( ]
z M z x ox
z M z ox
z M z x o
+
−
′∈ ⇔ ∈
∈ ⇔ ∈
′∈ ⇔ ∈
(c
( ) ( )( ) [ )( ) ( ]
z i M z y oy
z i M z oy
z i M z y o
+
−
′∈ ⇔ ∈
∈ ⇔ ∈
′∈ ⇔ ∈
2 (".
(a
( ) ( )( ) ( ) ( )
( ) ( )
aff M aff M M M
aff MM aff M aff M
MM aff M aff M
′ ′= ⇔ =
′ ′= −
′ ′= −
(b
( ) ( )( ) ( ) ( )( ) ( )
( )
aff u aff v u v
aff u v aff u aff v
aff u aff u
u aff u
α α
= ⇔ =
+ = +
=
=
(c G .! ( )( ) , ,A Bα β
( ) ( ) ( )( )1aff G aff A aff Bα β
α β= +
+
(d I- [ ]AB ( ) ( ) ( )( )1
2aff I aff A aff B= +
(e A B C % 78 9: , ,c B Az z z
A B≠ A B C )* , )* #
C A
B A
z z
z z
− ∈−
.
V( * & )+, - . 1 (* ( *z ∈ ( )M z %#z ; 8
'( )1,e OM
. / ' argz
(* ( ) [ ]1arg , 2z e OM π≡
#$% :
[ ][ ]
*
*
*
arg 0 2
arg 2
arg
z z
z z
z z k
π
π π
π
+
−
∈ ⇔ ≡
∈ ⇔ ≡
∈ ⇔ =
[ ]
[ ]
*
*
*
arg 22
arg 22
arg2
z i z
z i z
z i z k
π π
π π
π π
+
−
∈ ⇔ ≡
∈ ⇔ ≡ −
∈ ⇔ ≡ +
2 ( z *
( )cos sinz r iθ θ= + z r= [ ]arg 2z θ π≡ <)=
# %::z [ ],z r θ=" iz reθ=.
3 (#$%: a( [ ] [ ] [ ], , 2r r r r θو θ θ θ π′ ′ ′ ′= ⇔ = ≡
(b %:: z a ib= +% .
( )
2 2
2 2 2 2
2 2 2 2cos sin ,
a bz a ib a b i
a b a b
a b i a bθ θ θ
= + = + + + +
= + + = +
(c cos sin cos( ) sin( )i iα α α α− = − + −
cos sin cos( ) sin( )i iα α π α π α− + = − + −
cos sin cos( ) sin( )i iα α π α π α− − = + + +
sin cos cos( ) sin( )2 2
i iπ πα α α α+ = − + −
4 (
[ ] [ ] [ ][ ][ ][ ]
[ ][ ]
, , ,
, ,
,,
,
1 1,
,
, ,
n n
r r rr
r r n
r r
r r
r r
r r
θ θ θ θ
θ θ
θθ θ
θ
θθ
θ θ
′ ′ ′ ′⋅ = +
=
′= − ′ ′ ′
= −
= −
( ) [ ][ ]
[ ]
[ ]
( ) [ ]
arg arg arg 2
arg arg 2
arg arg arg 2
1arg arg 2
arg arg 2
n
zz z z
z n z
zz z
z
zz
z z
π
π
π
π
π
′ ′≡ +
≡
′≡ − ′
≡ −
≡ −
( )ii ie e e θ θθ θ ′+′⋅ = ( )ni ine eθ θ= i ie eθ θ−=
( )i
i
i
ee
e
θθ θ
θ′−
′ = 1 ii
ee
θθ
−=
1 ie π− = 2
ii e
π
= 2
ii e
π−− =
5( ( ) ( )( )[ ]( ) ( )( ) ( )( )[ ]
1, arg 2
, arg arg 2
e u aff u
u v aff v aff u
π
π
≡
≡ −
( ) ( )[ ]
( ) [ ]
1, arg 2
, arg 2
B A
D C
B A
e AB z z
z zAB CD
z z
π
π
≡ −
−≡ −
#$% :
)*[ ]1,C A
B A
z z
z zθ−
=−
( ) [ ]
1
, arg 0 2
C A C A
B A B A
C A
B A
z z z zACAC AB
AB z z z z
z zAB AC
z zπ
− −= = = ⇒ =
− −
−≡ ≡ −
6 ( ."Moivre
( ) ( ) ( )cos sin cos sinn
i n i nθ θ θ θ+ = +
7( ."ulerE
( )
( )
1cos
21
sin2
ix ix
ix ix
x e e
x e ei
−
−
= +
= −
2cos
2 sin
ix ix
ix ix
e e x
e e i x
−
−
+ =− =
#$% : ;4 7 >! %:: ?=
/1. :: @ # .
2 1i iz e z eβ α= =
( )1 2 cos sin cos sin
cos cos sin sin
2cos cos 2sin cos2 2 2 2
2cos cos sin2 2 2
i iz z e e i i
i
i
i
α β α α β βα β α β
α β α β α β α β
α β α β α β
+ = + = + + += + + +
+ − + −= +
− + + = +
( )1 2 cos cos sin sin
2sin sin 2cos sin2 2 2 2
2sin cos sin2 2 2
z z i
i
i
α β α βα β α β α β α β
α β α β α β
− = − + −+ − + −= − +
− + + = +
/2 . %#A ' #.
2 2 21 2
2 2cos2
i i ii i
i
z z e e e e e
e
α β α β α βα β
α β α β
+ − −−
+
+ = + = +
−=
2 2 21 2
2 2 sin2
i i ii i
i
z z e e e e e
e i
α β α β α βα β
α β α β
+ − −−
+
− = − = +
−=
VI(- * 01 . 1( *z ∈ *n ∈ % ! %#z z
nz Z=.
2( nz a= )! %= a.
3( [ ],Z r θ= *)! Z A %=
2, / 0,1,...., 1n
k
kz r k n
n n
θ π = + ∈ − = n
4( )!1 A %= 2
1,k
kw
n
π =
0,1,..., 1k n∈ −
5 ( 01*
a( / : [ ],Z r θ= *
)! Z .[ ]2
, ,2
Z r rθθ = =
)* )! Z= : ,2
u rθ =
u−
b(1 / : 1( + 02 *Z a += ∈
: ( )2
Z a a= =
)* )! Z= u a= u− 2( + 02 ( )*Z a a += − ∈
( ) ( )2 22Z a i a i a= − = =
)* )! Z= u i a= u− 33+ 02 ( )*Z ib b += ∈
( ) ( )2 2
22 . 1 1
2 2 2
b b bZ ib i i i
= = = + = +
)! )* Z = ( )12
bu i= + u−
4(+ 02 ( )*Z ib b += − ∈
( ) ( )2 2
22 . 1 1
2 2 2
b b bZ ib i i i
= − = − = − = −
)! )* Z = ( )12
bu i= − u−
5( + 02 Z a ib= + . ( )0 0b ≠aو ≠
): )! :
3 4Z i= − + .B z x iy= + 2 2 2 2z x y ixy= − +
2 2 2z x y= + 5Z =
( )( )( )
2 22
22
2 2
3 1
2 4 2
5 3
x yz Z
z Z xyz Z
x y
− = −= = ⇔ ⇔ = = + =
)1 () +3 (" E#22 2x =% 1x = " 1x = − )1 (–) 3 (" E#22 8y =
% 2 4y = " 2y = " 2y = − 9( )2 ( 2 0xy = ⟩)* y xوG ;4 7
)* 1
2
x
y
= =
" 1
2
x
y
= − = −
)! )* Z= 1 2u i= + u−
VII( 1 4 II: ":
2 0az bz c+ + = . 0a ≠ .B 2 4b ac∆ = −
1 H )* 0∆ = 9 +, :2
bz
a= −.
2H )* 0∆ ≠ +, ∆ )! u −uو
9 :2
b uz
a
− += 2
b uz
a
− −=.
#$%: (* 2 0az bz c+ + = . 0a ≠
)*2 1z ,+zو % :
1 2
1 2
bz z
ac
z za
− + = =
(* 2 2 0az b z c′+ + = . 0a ≠ ( ' # !"
b ac′ ′∆ = −
1H )* 0′∆ = 7 bz
a
′= −
2H )* 0′∆ ≠9 7 :
1 2
b uz
a
′− += 2 2
b uz
a
′− −=. u. )! ′∆.
I
1 !" (a f 0x
( ) ( )0
0
0
limx x
f x f xl
x x→
−= ∈
− ( )0f x l′ =.
(b f 0x
( ) ( )0
0
0
limx x
f x f xl
x x+→
−= ∈
− ( )0df x l′ =.
(c f 0x
( ) ( )0
0
0
limx x
f x f xl
x x−→
−= ∈
− ( )0gf x l′ =.
(d f 0x f 0x ( ) ( )0 0d gf x f x′ ′=.
2 # $ %. (a f 0x ! fc "
( )T ( )( )0 0,M x f x #$% ( )0f x′$ &#
( ) ( )( ) ( )0 0 0:T y f x x x f x′= − +. (b f 0x! fC '( "
)( )1T ( )( )0 0,M x f x$% # ( )0df x′ $ &# ( ) ( )( ) ( )1 0 0 0: dT y f x x x f x′= − +.
(c * % & .
(d ( ) ( )0
0
0
limx x
f x f x
x x+→
−= +∞
−! f +
0x fC $% , -. ) '( " , ( )( )0 0,A x f x.
(e ( ) ( )0
0
0
limx x
f x f x
x x+→
−= −∞
−! f +
0x fC $% , -. ) '( " "/, ( )( )0 0,A x f x.
(f ( ) ( )0
0
0
limx x
f x f x
x x−→
−= +∞
−! f +
0x fC $% , -. ) '( " "/, ( )( )0 0,A x f x.
(g ( ) ( )0
0
0
limx x
f x f x
x x−→
−= −∞
−! f +
0x fC $% , -. ) '( " , ( )( )0 0,A x f x.
&'(: (* f 0x !fC "
-&( )( )0 0,M x f x) 3.( (* f + 0x ! fC
) ( ( )( )0 0,M x f x. .
3 *+ f I g ( )f I
!g fο I
( ) ( ) ( ) ( )( ) ( )x I g f x g f x f xο ′ ′ ′∀ ∈ = ⋅
4 +" f"% # I
( ) ( ): 0x I f x′∀ ∈ ≠! 1f − ( )J f I=
( ) ( ) ( ) ( )( )1
1
1:x J f x
f f x−
−′∀ ∈ =
′ .
5 ," .
&'(: (a u"% & I.
& ( ) ( )nf x u x= ( ) / 0fD x u x− =.
(b f 4( 4 & 0x & 5 6&# % &% 0x . & %f 0x
4 "&# "#.
1 (
( ) ( ) 0a a ′∈ = 12 (( )f g f g′ ′ ′+ = +
2 (( ) 1x ′ = 13 (( )af af′ ′=
3 (( )ax a′ = 14( ( )f g f g fg′ ′ ′⋅ = + 4 (
( ) 1r rr x rx −′∈ = 15 (( ) 1r rf rf f −′ ′= ⋅
5 (2
1 1x x
′ = −
16 (2
f f g g f
g g
′ ′ ′ −=
6 (( ) 1
2x
x
′ = 17 (2
1 f
f f
′ ′ −=
7 (( )( ) ( )( )2
u xu x
u x
′′ = 18( ( )sin cosx x′ =
8 (( )( ) ( )( )( )
32
33
u xu x
u x
′′ = 19 (( )cos sinx x′ = −
9 (( )( ) ( )( )( ) 1
nn
n
u xu x
n u x−
′′ = 20 (( ) 22
1tan 1 tan
cosx x
x′ = + =
10 (( ) 2
1tan
1Arc x
x′ =
+
21(
( )( )( ) ( ) ( )( )sin cosu x u x u x′ ′=
11 (
( )( )( ) ( )( )( )2tan
1
u xArc u x
u x
′′ =+
22 (
( )( ) ( ) ( )( )cos sinu x u x u x′ ′= −
23 (( )( )( ) ( ) ( )( )( )2tan 1 tanu x u x u x′ ′= +
6 : f"% & I.
(a f&. I : ( ) ( ) 0x I f x′∀ ∈ ≥.
(b f ( I : ( ) ( ) 0x I f x′∀ ∈ ≤.
(c f I : ( ) ( ) 0x I f x′∀ ∈ =.
7 !-: f"% & I . & f ′′ 0x & f ′ 89 4 6&# 0x.
8 ": f"% & I.
(a fC & "∪ " ( ) ( ): 0x I f x′′∀ ∈ ≥. (b fC #" ∩" ( ) ( ): 0x I f x′′∀ ∈ ≤.
9 !-" -. f"% & I 0x I∈ .
( )( )0 0,M x f x '# f ′′ 89 4 6�x.
&'(: (a f 89 4 3 6&# 0x !
( )( )0 0,M x f x ;< ) '# "(, .
(b # & 5 '# =% && &5 ( )f x′′> ?& .
II # 1 '– /+ . (a6 ( ).x a∆ = "* fC : (*" x fD 2 fa x D− ∈
(*( ) ( ) ( ): 2fx D f a x f x∀ ∈ − =
(b ( ),a bΩ "* . fC : (*" x fD 2a x D− ∈
(*( ) ( ) ( ): 2 2fx D f a x b f x∀ ∈ − = −
2 0$( 1 . a(!"
" fC @> 3 " ( )limx a
f x→
= ∞
5( )limx
f x b→∞
=5 ( )limx
f x→∞
= ∞
b( 0$( 1 !2 1 ( ( )lim
x af x
→= ∞
!6 ( ) : x a∆ =" fC% a. 2 ( ( )lim
xf x b
→∞=
! 6 ( ) : y a∆ =" fC% ∞.
3 ( ( )limx
f x→∞
= ∞ 6 ( )limx
f x
x→∞.
(a ( )limx
f x
x→∞= ∞
!fC% , $<% % " ∞.
(b ( )lim 0x
f x
x→∞=
!fC% "(, $<% % " ∞.
(c ( )lim 0x
f xa
x→∞= ≠ 6 ( )( )lim
xf x ax
→∞−.
(i ( )( )lim f x ax b− =
! 6 ( ) : y ax b∆ = +" fC% ∞. (ii ( )( )lim f x ax− = ∞
! fC$<% % " y ax=% ∞.
&'(:
6 ( ) : y ax b∆ = +" fC% ∞ ( ) ( )( )lim 0f x ax b− + =
5 < "A (B ;< "#( ) : y ax b∆ = + 5 ( )f x" ( ) ( )f x ax b h x= + +
=( )lim 0x
h x→∞
=.
3 &'( ,". (a &# " ( )f x m== "(5 < fC =
6 ( ) : y m∆ =. (b &# " ( ) 0f x == "(5 < fC = ,"(. (c &# " ( ) ( )f x g x== "(5 < fC gC. (d % " ( ) ( )f x g x≤ 3% < >fC gC. (e=C & "%5 fC6 ( ) : y ax b∆ = + 6
8 &( )f x y− (* ( ) 0f x y− ≥! fC &% ( )∆. (* ( ) 0f x y− ≤! fC &% ( )∆.
.
I .
( ), , ,o i j k
.
1 ! ( ) ( ), , , ,v x y z u x y z′ ′ ′
2 2 2 .u x y z u v xx yy zz′ ′ ′= + + = + +
2 "# ! ( ) ( ), , , ,B B B A A AB x y z Aو x x z
( ), ; ;B A B A B AAB x x y y z z− − −
( ) ( ) ( )2 2 2
B A B A B AAB x x y y z z= − + − + −
3 ! ".
(a ( )P$ . % & & ( )P & n
# & ( )D % '% ( )P.
(b $ ! ( ) : 0P ax by cz d+ + + =
& ( ), ,n a b c
% & ( )P.
c(# $ %" &'( ) ( .
( : $ & & *( )P ( )1, 1,2A −
& ( )2,1, 1n −+% & :
"'%1 : ( ) ( ), , . 0M x y z P AM n∈ ⇔ =
( ) ( ) ( )
1 2
1 . 1 0
2 1
2 1 1 2 0
x
y
z
x y z
− ⇔ + = − −
⇔ − + + − − =
, : ( ) : 2 1 0P x y z+ − + =
"'%2 : ( )2,1, 1n − % & ( )P & , ( )P %
-2 1 0x y z+ − + = ( ) ( )1, 1,2A P− ∈ ,
2 1 2 0d− − + = 1d = , ( ) : 2 1 0P x y z+ − + =.
d(" ( .
( ) ( )D # ′Dو ( , , )u a b c
( ', ', ')v a b c
% :
( ) ( )D D′⊥ , "# , u v⊥ . 0u v =
0aa bb cc′ ′ ′+ + =. e() *" ( .
(i ( )D + # u
( )P + $ w vو
.
( ) ( )D P⊥ , "# ,
u v
u w
⊥ ⊥
. 0
. 0
u v
u w
= =
.
(ii ( )D + # ( , , )u a b c
( )P .*! $
( , , )n α β γ+% & .
(* ( ) ( )D P⊥ 1 , "# , n uو
# .
0a a b
b c c
α α ββ γ γ
= = =
(* ( ) ( )//D P , 1 , "#u n⊥ . 0u n =
(iii # , ( )D $ % '% ( )P2 :
(* & & ( )D % & ( )P.
(* % & & ( )P & ( )D.
f( ( .
(i ( ) ( )Q Pو n
n ′ % % .
(* ( ) ( )P Q⊥ , "# , n n′⊥ . 0n n′ =
.
(* ( ) ( )//P Q , "# , n
n ′# .
(ii ! ( ) : 0P ax by cz d+ + + =
( ) : 0Q a x b y c z d′ ′ ′ ′+ + + =
( ) ( )P Q⊥ , "# , 0aa bb cc′ ′ ′+ + =.
g() %" ! .
(i ( )P $ A &"#
H ' "# A % ( )P
& AH & A %( )P
( )( ),d A P AH=.
(ii $ ! ( ) : 0P ax by cz d+ + + =
&"# ( )0 0 0, ,A x y z.
( )( ) 0 0 0
2 2 2,
ax by cz dd A P
a b c
+ + +=
+ +.
(II . 1 34 & Ω %- r 3 "# &% M
5#*M rΩ =.
2 & & ( )S 34 ( ), ,a b cΩ %- r3 :
( ) ( ) ( )2 2 2 2x a y b z c r− + − + − =
! #- % & - :2 2 2 0x y z x y zα β γ δ+ + + + + + =.
n
)P
( )D
n
)P .A .M
n
)P
u
( )D
n
)P
( )D
)P H
A
3 &% !
( ) 2 2 2: 0x y z x y zα β γ δΓ + + + + + + =
&% &!" & ( )Γ#" 63 .
"'%1 : 72 2 2
d c b aγ β αδ − − −= = = = *
2 2 2a b c d+ + −
(* ,2 2 2 0a b c d+ + − < 2 Γ = ∅
(* ,2 2 2 0a b c d+ + − = 2 ( ) , ,a b cΓ = Ω
(* ,2 2 2 0a b c d+ + − > 34 & 2 ( ), ,a b cΩ
%-2 2 2r a b c= + +. "'%2 - % & *! #
( ) ( ) ( )2 2 2x a y b z c k− + − + − =
&3 &#!" &! !
2 22
2 2X X X
α αα + = + −
(* ,0k < 2 ( )Γ = ∅
(*, 0k = 2 ( ) ( ) , ,a b cΓ = Ω
(* ,0k > 2 ( ), ,a b cΩ %- r k=.
4 +'%, -. !'( ! (.
( )S * & 3"8[ ]AB & % 9* ( )S 63
#": "'%1
( ) ( ), , . 0 . 0A B
A B
A B
x x x x
M x y z S AM BM y y y y
z z z z
− − ∈ ⇔ = ⇔ − − = − −
( )( ) ( )( ) ( )( ) 0A B A B A Bx x x x y y y y z z z z⇔ − − + − − + − − =
"'%2: &:9 ;-!
( )( ) ( )( ) ( )( ) 0A B A B A Bx x x x y y y y z z z z− − + − − + − − =
5 ) ! 0%".
(a ( )S 34 & Ω %- r ( )P & $
7"#( )S ( )P *! # ( )( ),d d P= Ω1<* .=( 63 :
(i 1 , d r> 2 ( )P >? ( )S) ( )P 7"# < ( )S.(
(ii , d r= 2 ( )P ( )S ;* &"# "# H #
& * A,3 ( )P B ( )S H B "# H 3 "#
' Ω % ( )P.
(iii 1 , d r< $ 2 ( )P 7"# ( )S ;C 5 ( ) $ ; ( )P 3 34 H ' "# Ω
%( )P %- 2 2r r d′ = −.
(b 1 , ( )( ),d PΩ 2 ( )PΩ ∈# $ & * A,3
( )P'"8 $ . $ & * A,3 ( )P & 7"# ( )S 5
$! ;C ( ) ; ( )P34 Ω 3 %-
r.
(c ( )S 34 & Ω %- x.
(i ( )P ( )S, , "# ( )( ),d P rΩ =.
(ii ( )P ( )S A , "# , ( )AΩ % '%
( )P A.
(iii & B $ ( )S A $ 3 A AΩ
+% &. 6 *" ! 0%":
# ! ( )0
0
0
:
x x t
D y y t
z z t
αβγ
= + = + = +
& ( )S :
( ) 2 2 2: 0S x y z ax by cz d+ + + + + + =
& 7"# & ( )S # ( )D& *! # :
( )( )( )
( )
0
0
02 2 2
1
2
3
0 4
x x t
y y t
z z t
x y z ax by cz d
αβγ
= + = + = + + + + + + + =
, x y z )4 ( &( & & % 9* t.
∆& A,3 4 :
(i , 0∆ < B & 2 , * ( )D 7"# < ( )S.
(ii , 0∆ = , * =* !# & 2 ( )D 7"# ( )S
;* &"#H # & * A,3 ( )D B ( )S H.
(iii , 0∆ > ? * !# & 2 2 1t ) , tو )D 7"#
( )S "# B *% 9 A 1(*و B Aو
2 1t tو )3( ) 2( ) 1.(
(III
1D ( ), , ,o i j k
-! .
!
x
v y
z
′ ′ ′
x
u y
z
.y y x x x x
u v i j kz z z z y y
′ ′ ′∧ = − +
′ ′ ′
2 u
v
, "# , # 0u v∧ =
3 (a 1 , u
v
$ ( )P 2 u v∧
% &( )P.
(b , ,C B A &# E "# .=( ) 0AB AC∧ ≠
( &
AB AC∧
$ % & ( )ABC.
4 .( &* ( )ABC 3 1
2S AB AC= ∧
5 F= '4 &*( )ABCD 3 S AB AD= ∧
6 ( )D # A & ! + u
M&"# .
( )( ),AM u
d M Du
∧=
7 (* u v v u∧ = − ∧
(* ( ) ( )u v u v u vα α α∧ = ∧ = ∧
(* ( )u v w u v u w∧ + = ∧ + ∧
(* ( )u v w u v v w+ ∧ = ∧ + ∧
8 # 7"# & ( )D & ( )S *
( )( ),d DΩ ( 7"# >.
وا یراوال ا
(I 1
F
] [: 0,
1
f IR
xx
+∞ →
→
(1) 0F = ln Log
(a (∗ ] [ln : 0, IR+∞ →
(∗ ] [ln 0,D = +∞
(b (∗ ( ) ln( ( ))f x u x=
( ) 0fx D u x∈ ⇔ >
(∗ ( ) ln ( )f x u x=
( ) 0fx D u x∈ ⇔ ≠
(c (∗ ln(1) 0= (∗ ln( ) 1e = ) 2,71828e (
(∗ ( ) : ln( )rr Q e r∀ ∈ = 2 ln
0a > 0b > r Q∈ (∗ ln( ) ln lnab a b= +
(∗ ln( ) lnra r a=
(∗ 1
ln( ) ln aa
= −
(∗ ln( ) ln lna
a bb
= −
(∗ ln lna b a b= ⇔ = (∗ ln lna b a b< ⇔ <
: (∗ !" 0ab > #$ :ln( ) ln lnab a b= +
(∗ !" 0a
b> #$ :ln( ) ln ln
aa b
b= −
(∗ !" 0na > #$ :ln( ) lnna n a=
3 ln( )x
4
1( 0) : (ln ) ' (x x
x∀ > = ∗
'( )(ln ( )) ' (
( )
u xu x
u x= ∗
'( )(ln ) ' (
( )
u xu x
u x= ∗
5 ! " lim ln( ) (
xx a
→+∞= +∞ ) ln( )+∞ = +∞ (
0
lim ln( ) (x
x b+→
= −∞ ) ln(0) = −∞ (
(c lnlim 0x
x
x→+∞=
(d 0
lim ln 0x
x x+→
=
(e 0
ln(1 )lim 1x
x
x→
+ = 1
lnlim 1
1x
x
x→=
−
ln( )
( ) ln( ( ))( ) 0 ln
( )ln
( ) 01
tv x
tu x v x
v x t tw x
tv x a
t
+
→ +∞ →
→ →
→ ≠ →−
(II #$ 1 & ln
xx e→
(a (∗ xx e→ '( $ R.
(∗ ( ) : 0xx e∀ ∈ >R
(b (∗ 0 1e = (∗ 1e e=
(c (∗ ( ) : ln( )xx e x∀ ∈ =R
(∗ ln( )( 0) : xx e x∀ > =
(d (∗ ( ) ( 0) : ( )xx y e y ln y x∀ ∈ ∀ > = ⇔ =R
2 ,x y ∈R r Q∈
(∗ x y x ye e e+ = ⋅ (∗ 1x
xe
e− =
(∗ x
x yy
ee
e− = (∗ ( )rx x re e=
(∗ x ye e x y= ⇔ =
(∗ x ye e x y< ⇔ <
3
(∗ ( ) : ( ) 'x xx e e∀ ∈ =R
(∗ ( ) ( )( ) ' '( )u x u xe u x e=
x
ln x
1 0
0 + + −
+∞
4 ! " lim (x
xe a
→+∞= +∞ ( )e+∞ = +∞
lim 0 (x
xe b
→−∞= ( 0 )e−∞ =
lim (x
x
ec
x→+∞= +∞
lim 0 (x
xxe d
→−∞=
0
1lim 1 (
x
x
ee
x→
− =
( )
( )
( ) ( )( )
( )1
( )
t
v xt
t
ev x
tu x e x
v x tew x
ev x a
t
ϕ→ +∞ →
− → −∞ →−→ ∈ →R
(III #%& a .
1
1a ∗+∈ −R &* a
loga $ :
ln( 0) : log ( )
lna
xx x
a∀ > =
( (* 10log+ '
log
10
ln( 0) : log( ) log ( )
ln10
xx x x∀ > = =
(* & ,! - e. 2 (∗ loga ,. &/ ln.
(∗ log ( ) 1a a = (∗ log (1) 0a =
(∗ log(10) 1= (∗ log(1) 0=
(IV )* +, - - #$ 1
ln( )( 0)( ): x x aa x a e∀ > ∀ ∈ =
2 0a > 0b > x y R .
(∗ x y x ya a a+ = ⋅ (∗ ( )xy x ya a=
(∗ x
x yy
aa
a− = (∗ ( )x x xa b ab=
(∗ 1x
xa
a− = (∗ ( )
xx
x
a a
b b=
(∗ x ya a x y= ⇔ =
(∗ ln( ) lnxa x a=
ل ا
(I
1(
E ( )card E
2(
n ! . # n $%& & !n #' ( )*& :
(∗ ! 1.2.3.......n n= ,- &./ 0n ≠.
(∗ 0! 1=.
3( .
0 ,1 # ,- &./p ' ,- &2& :
(∗ 1n 34 25 6'! 1 .
(∗ 2n 34 25 6'! 2.
(∗ pn34 25 6'! p.
7&28& 9% : 3; <& =>& ,?@1 2. ...... pn n n.
4( – "# – $
1 2, ,..., nE a a a= n A . p n≤.
(a B ;; p CD A n E ;& ;; 1 p E
E;; F- Bp G#H A E . BD ; :1 2( , ,...., )nx x x
(b BD $%& & 7 ;& 9% :p
nA& #'( )* :
!( 1)( 2).......( 1)
( )!p
n
n facteurs
nA n n n n p
n p= = − − − +
−
(c #' ; E B ;; F- n CD A n E
(d 7I' & 9% ( 1)( 2)......1 !nnA n n n n= − − =
(e B JK; p CD A n E ;& JK; 1 p E
, LM F-p G#H A E . BD JK : 1 2, ,...., px x x
(f BD $%& & 7JK& 9% :p
nC #'( )*& :
! ( 1)( 2).......( 1)
! !( )! ( 1)( 2).....................1
p facteurs
pp n
n
A n n n n n pC
p p n p p p p
− − − += = =− − −
5( %&
(a 0 1nn nC C= =
1 1nn nC C n−= =
p n pn nC C −=
1 11
p p pn n nC C C+ +
++ = .
(b &N& OA& .0
( )n
n k n k kn
k
a b C a b−
=
+ =∑
(c E n A . P&M1 E 2n.
(II ' ()
1( *# .
Q& R 1 2, ,..., na a aΩ = ) 7T& ,- (
V# WX& @ 4 / Y6Ω A F- >D &./ Z6@ &./ ia Ω D
66Xip [\ :(∗ 0 1ip≤ ≤
(∗ 1 2 1np p p+ + + =
E ( )i ip a p= . ]& ( , )pΩ X& &P^@ V ' .
2( +( ' ( :
( , )pΩ X& &P^@ V ' A _X .
`N& YX&AT& `&X8& 7WX& a ; <& &D . b' .
,- &./ 1 2, ,..., nA a a a= ,?@ 1 2( ) ( ) ( ) ( )np A p a p a p a= + +
3(%& ( , )pΩ X& &P^@ V ' .
(a A B C_X [ \A B = ∅∩ '
( ) ( ) ( )p A B p A p B= +∪
(b A B C_X . '( ) ( ) ( ) ( )p A B p A p B p A B= + −∪ ∩
(c A _X AN& B ^*& `A ' ( ) 1 ( )p A p A= −
(d 1A 2A ..... nA ' cd cd #AJ _&X1
1 2 1 2( .... ) ( ) ( ) ( )n np A A A p A p A p A= + + +∪ ∪ ∪
4( - . () /0 1
( , )pΩ YXT& eJ 7T1 fg ,' [\ X& &P^@ V '
(∗ YXT& eJ h &DT& `&X8& fi 1
( )card Ω
(∗ A _X . '( )
( )( )
card Ap A
card=
Ω
2(": (a ,?@ YXT& eJ &DT& `&X8& fg ,- &./
( )nombredecas favorables
p Anombredecas possibles
=
(b 1 j k D l& m n; ,1 o 7WXT& $ ; p@ ,/ kq rs 6'>D
#' - ) : tuO rs –quO rs 6 >4 w e#D D & o W 7&-(
(c 7>*& YxD &DT& `&X8& YX& X W1 Ey quO D0& z- &./
& Yx& |O& # YX : ,- &./ 1 2, ,..., na a aΩ = ,?@
1 2( ) ( ) ( ) ( )np A p a p a p a= + +
'3 : 4 & M 1 / 6uO eJ h M& ~48& [\ t
T& $@ 34 YX& G^ M 34 YX& YXT& eJ h DJ& ~48& YX
`N& YX& E X1A" N& B G^ 34 V# YA3"
45: ' 1,2,3,4,5,6Ω =
(∗ &DT& `&X8& YX& E .
f^ (1) (3) (5)p p p x= = = (2) (4) (6) 2p p p x= = =
'(1) (2) (3) (4) (5) (6) 1p p p p p p+ + + + + =
b' 2 2 2 1x x x x x x+ + + + + = b' 1
9x = ,./
1(1) (3) (5)
9p p p= = =
2(2) (4) (6)
9p p p= = =
(∗ ' 3,6A = ,./ 1 2 1
( ) (3) (6)9 9 3
p A p p= + = + =
5( 6 ' () :
A B [\ C_X ( ) 0p A ≠
`N& YX1B `N& ,1 # A 6 ( )
( / )( )
p A Bp B A
p A= ∩
6( 7 8 . () 9%
A B [\ C_X ( ) 0p A ≠
( ) ( ) ( / )p A B p A p B A∩ =
7( : . () 9%
(a`&X8& ,/ Y6 1A 2A ..... nA B ' ,;Ω ,- &./ Z6@ &./
(∗ ( ) : i ji j A A∀ ≠ = ∅∩ (∗ 1
n
ii
A=
= Ω∪.
(b`&X8& ,; 1A 2A ..... nA B ' Ω #AJ z- &./ Z6@ &./
*& `&X8& ,; cd cd .
c( #& 7WXT& O
1A 2A ..... nA B ' ,; _&X1 Ω. _X FB ' :
1 1( ) ( ) ( / ) ( ) ( / )n np B p A p B A p A p B A= + +
8("; <)
(a C_N& ,/ Y6 A B ,- &./ Z6@ &./ ,I6
( ) ( ). ( )p A B p A p B=∩
(b ,_N& ,' A B ,- &./ Z6@ &./ C#6 ( / ) ( )p B A p B=
( / ) ( )p A B p A=2& V# _' W X1 6 ,- &./ b' .
(c D R n cd cd #6 2& .
A X& 2& m 66 YX& _X ( )p A p=
B `N& ": `N&A 6' k YI2 Z ^D k n 2& "
' :( ) ( ( )) (1 ( ))k k n knp B C p A p A −= −
2(" #' f YX& X FM1 JAD :
(a F & 2T& 1 & E & ' ,- &./ p
nC ( )( )
( )
card Ap A
card=
Ω
(b ,' 7& 2& k / D0& 9% J 7& 2& k D z- &./
76 >& 7 ;& Yx& E0 X & @& @ . BD / F :
1 2( , ,...., )nx x x [X ix 34 D0& 0 i .
(III 6 =9 8 .
(1 >; F- &u rO X / F- ZD' Ω 66X D
%2K' <& 36# rO*&X BD ( )X Ω .
(2 X [\ &u rO 1 2( ) , ,...., nX x x xΩ =
FX& ,4 X 4 / F6X \ 4 &./ ( )ip X x= F
1,2,...,i n∈ #' - YM m & 9% l# :
nx . . . . . . . . . . . . . . 2x 1x ix
nα . . . . . . . . . . . . . . 2α 1α ( )ip X x=
3( 1# 4> .
&u rO* p'& F8& X BD $%& & ( )E X#' ( )*& :
1 1 2 2
1 1 2 2
( ) ( ) ( ) ( )n n
n n
E X x p X x x p X x x p X x
x x xα α α= = + = + + == + + +
4(? #98
&u rO* k'O*&X BD $%& & ( )V X #' ( )*& :
2 2( ) ( ) ( ( ))V X E X E X= − [X
2 2 21 1 2 2
2 21 1 2 2
( ) ( ) ( ) ( )nn n
nn n
E X x p X x x p X x x p X x
x x xα α α= = + = + + =
+ + +
5( /@ A B C) .
& &u rO* $&>& )&TX BD $%& & ( )Xσ#' ( )*& :
( ) ( )X V Xσ =
6( D#EF
&u& rO# '0& & X <& && BD h F #' ( @*& :
( ): ( ) ( )x F X p X x∀ ∈ = <
X 4 / Y6 &&F \ 4 &./ ( )F x F x .
'3 : =A& R3
4
BU
N ; E 3 =AA& 7&- .
X # FA& P^ & 7&& D x F- ZD' $%& &u& rO*& .
(a =9 8 GH&I J K;X :
(∗ 0X = V# YAN& b' 3N.
(∗ 1X = V# YAN& b' 1 , 2B N .
(∗ 2X = V# YAN& b' 2 ,1B N .
(∗ 3X = V# YAN& b' 3B .
,./ ( ) 0,1, 2,3X Ω =
b( ' ( LM X . 3437
4( 0) (
35
Cp X
C= = = ∗
1 23 4
37
18( 1) (
35
C Cp X
C= = = ∗
2 13 4
37
12( 2) (
35
C Cp X
C= = = ∗
3337
1( 3) (
35
Cp X
C= = = ∗
3 2 1 0 ix
1
35
18
35
18
35
4
35
( )ip X x=
(c 4 4 18 1 49
( ) 0. 1. 2. 3.35 35 35 35 35
E X = + + + =
(d '2 2 2 2 24 4 18 1 75
( ) 0 . 1 . 2 . 3 .35 35 35 35 35
E X = + + + =
,./2 2 275 49 224
( ) ( ) ( ( )) ( )35 35 352
V X E X E X= − = − =
(e ' 224
( ) ( )35
X V Xσ = =
(f '0& & . E ( )F x F x .
(∗ ,- &./ 0x ≤ ,?@ ( ) ( ) ( ) 0F x p X x p= < = ∅ =
(∗ ,- &./ 0 1x< ≤ ,?@ 4
( ) ( ) ( 0)35
F x p X x p X= < = = =
(∗ ,- &./ 1 2x< ≤,?@
22( ) ( ) ( 0) ( 1)
35F x p X x p X p X= < = = + = =
(∗ ,- &./ 2 3x< ≤ ,?@
34( ) ( ) ( 0) ( 1) ( 2)
35F x p X x p X p X p X= < = = + = + = =
(∗ ,- &./ 3 x< ,?@
( ) ( ) ( 0) ( 1) ( 2) ( 3) 1F x p X x p X p X p X p X= < = = + = + = + = =
ــــــــــــــــــــــ ا
(Iــــــــــیــــــــ ت . f . I b و a و Iل دا
ـ b إ a f ت ) ا#"د ا!ي )b
af x dx∫
): وا#ف ی ) ( ) ( )b
af x dx F b F a= −∫
()F "ا أص دا f . .
(* , [ ]( ) ( ) ( ) ( )b b
aaf x dx F x F b F a= = −∫
2ي - 01 x ی ت#/ی. ا- *)
( ) ( )b b
a af x dx f t dt=∫ ∫
(IIخــــ ـــــت f و g I c و b و a و Iل دا
(1 ( ) 0a
af x dx =∫
(2 ( ) ( )a b
b af x dx f x dx= −∫ ∫
(3 ( ) ( ) ( )b c b
a a cf x dx f x dx f x dx= +∫ ∫ ) 56 4ل ( ∫
(4 ( ( ) ( )) ( ) ( )b b b
a a af x g x dx f x dx g x dx+ = +∫ ∫ ∫
(5 ( ) ( )b b
a aaf x dx a f x dx=∫ ∫) (a IR∈ .
) ا"ا 6) ) ( )x
aF x f x dx= ا f ه ا"ا ا8ص "ا∫
. aت>#"م :
(7 (a إذا آن a b≤ و [ ]( , ) : ( ) 0x a b f x∀ ∈ ≥
) :@ن ) 0b
af x dx ≥∫
(b إذا آنa b≤ و [ ]( , ) : ( ) 0x a b f x∀ ∈ ≤
) :@ن ) 0b
af x dx ≤∫
(c إذا آن a b≤ و [ ]( , ) : ( ) ( )x a b f x g x∀ ∈ ≤
) :@ن ) ( )b b
a af x dx g x dx≤∫ ∫
(d إذا آن a b≤ ن و@: ( ) ( )b b
a af x dx f x dx≤∫ ∫
(8 (aا#"د 1
( )b
af x dx
b a− f ی اC ا/سA "ا ∫
a و b (b ی/ج" "د c /ر F a و b (F
( ) ( ) ( )b
af x dx b a f c= −∫
(c 1
( )b
am f x dx M
b a≤ ≤
− ∫ ()m و M Cا Gه
/ی "ا Cا Cوا ا"/یf [ ],a b.
. J b GI و a تت, (8): اHص ــــــــ
(III ب ا" ـــــــــــ ت$#ت . . ا* ()'&اء )1
f و g قC4L F) ت/ن Iل دا 5 'f و
'g I و a و b I . <ی" :
[ ]'( ) ( ) ( ) ( ) ( ) '( )b bb
aa af x g x dx f x g x f x g x dx= −∫ ∫
IV( ول اوال ا) ا.-دی'
F f 0 1
0a ≠ ax
( 1)
rx
r ≠ − 11
1rx
r+
+
'
( 1)
ru u
r ≠ − 11
1ru
r+
+
2
1
x
1
x−
'u
u
1
u−
1
x 2 x
'u
u 2 u
1
x ln x
'u
u ln u
xe xe
axe 1 axea
F f
( )' u xu e ( )u xe
2
1
1 x+ arctan( )x
2
'( )
1 ( ( ))
u x
u x+ arctan( ( ))u x
cosx sinx sinx cosx−
2
2
1 tan
1
cos
x
x
+
= tanx
cos( )ax b+ 1 sin( )ax ba
+
sin( )ax b+ 1cos( )ax b
a− +
2
2
1 tan (
1
cos ( )
ax b
ax b
+ +
=+
1 tan( )ax ba
+
'( ) cos( ( ))u x u x sin ( )u x
'( )sin( ( ))u x u x cos ( )u x− 2'( )(1 tan ( ( ))u x u x+ tan ( )u x
(V ) ت ـــــــــــــ1 ا$#ـــــ
(1 ( )P x
I dxax b
=) ي 5 ∫+ )P x ax b+
# GU'u
u.
(2 2
1I dx
ax bx c=
+ +∫
(a 0 إذا آن∆ < /Cا Vد ا"F 2( )p x ax bx c= + +
F21 ( ( ))
I dxu x
α=+∫
OPو ( )t u x=
(b 0 إذا آن∆ > # ( )P x ( ) ( )( )p x a x xα β= − − GU
>W أن 1 1 1 1
( )( )P x x xα β α β
= −− − −
# GU 'u
u
(c 0 إذا آن∆ =
2 2
1 ( ) ' 1
( ) ( )
xI dx dx
x x x
αα α α
− = = = − − − − ∫ ∫
(3 2
( )P xI dx
ax bx c=
+ ) ي 5 ∫+ )P x
2ax bx c+ + # GU 'u
u أو
2
'
1 ( )
u
u+ .
(4 ( )nI P x ax bdx= أو ∫+( )
n
P xI dx
ax b=
+∫
OP nt ax b= +
(5 ( )cos( )I P x ax dx= ) أو ∫ )sin( )I P x ax dx= أو ∫
( ) kxI P x e dx= ∫ ← OPاء و ا 8ج
( ) ( )
'( ) cos( ).....
f x P x
g x ax
= =
(6 ( )cos ln( )I P x x dx= ) أو ∫ ) tanI P x Arc xdx= ∫
← OPاء و ا 8ج( ) ln ( arctan )
'( ) ( )
f x x ou
g x P x
= =
(7 cos( )kxI e ax dx= ∫ sin( )kxI e ax dx= ∫
I ا 8جاء ت و" ← A Iα= +
(8 1
xI dx
ae b=
+∫ .
'
( )
x x
x x x
e e uI dx dx
e ae b a be u
− −
− −= = =+ +∫ ∫ ∫
(9 (ln )rx
I dxx
= ∫
1(ln ) 1(ln ) '(ln ) (ln )
1
rr rx
I dx x x dx xx r
+ = = = + ∫ ∫
(10 ( ) ( )
( ( ))n
u x v xI dx
w x= ∫
← OPاء و ا 8ج
'( )'( )
( ( ))
( ) ....
n
w xf x
w x
g x
= =
(V$23ـــــ ت" ــــــــــــــب اـــــــت "ــــــب ا*"ــــت )1
(a f ] دا ],a b ( )a b< و ( )E Fا
/ر ـ FاfC و ( ' )x Ox و x a= و x b= .
Fا )( )Eه A(E)=( ( ) ) .b
af x dx u a∫
:4ـــــــ (* Y0 إذا آf ی/ج" :/ق F/رfC ی#> ≤
)=A(E):@ن ا8:ص ( ) ) .b
af x dx u a∫
(* Y0 إذا آf ی/ج" ت YFF/رfC ی#> ≥
)=A(E) ا8:ص :@ن ( ) ) .b
af x dx u a−∫.
إذا آY ت- ا\4رة 6Z :@ن *)
A(E)= ( ) ( )b
af x dx f x dx
α
α−∫ ∫.
(b f وg ] دا ],a b ( )a b< و ( )E
/ر ـ Fا FاfCو gC و x a= و x b= .
Fا )( )E ه A(E)=( ( ) ( ) ) .b
af x g x dx u a−∫
ـــــــ : (* Yإذا آ f g≥ <#ی fC ی/ج" :/ق gC
=A(E) :@ن ( ( ) ( ))b
af x g x dx−∫
(* Yإذا آ f g≤ <#ی fC YFی/ج" ت gC
=A(E) :@ن ( ( ) ( ))b
ag x f x dx−∫
(* Oإذا آن وض fC ـ N< gC-ی
@ن :
A(E)= ( ( ) ( )) ( ( ) ( ))b
ag x f x dx f x g x dx
α
α− + −∫ ∫
(c إذا آن i cmα=
jو cmβ=
u.2 :@ن و("ة N5س ا(ت ه/ a cmαβ=
9م ـــــــــ "ب ا67)2(a ( )S ) Vا أ` (
/ر ـ V و Fئ ا (G ا
( )S و ا/ی z a= وz b=
إذا آY ا"ا [ ]: ,
( )
S a b IR
t S t
→→
[ ],a b ن@: V=( ( ) ) .b
aS t dx u v∫
)( )S t ObCئ ت) ه ( ا )S و ا/ى z t= (
(b f ] دا ],a b
(/ل F/ر ا8:ص دورةfCإذا دار
آ :@ ی/" ی G دوران ، Gه!ا ا G)ه/ و
2( ( ( ) ) ) .b
aV f x dx u vπ= ∫