اﻟــــــﺪوال اﻷﺻـﻠﻴـــــﺔ -1ﺗﻌﺮﻳـــﻒ : ﻟﺘﻜﻦ fداﻟﺔ ﻋﺪدﻳﺔ ﻣﻌﺮﻓﺔ ﻋﻠﻰ ﻣﺠﺎل . Iﻧﻘﻮل أن اﻟﺪاﻟﺔ Fداﻟﺔ أﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ Iإذا وﻓﻘﻂ إذا آﺎن : Fداﻟﺔ ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق ﻋﻠﻰ اﻟﻤﺠﺎل . I )F '( x) = f ( x وﻟﻜﻞ xﻣﻦ : I ﻣﺜــــﺎل : -1ﻟﺘﻜﻦ إذن :
F ( x ) = x2 + x + 1 F '( x) = 2 x + 1
إذن :اﻟﺪاﻟﺔ Fهﻲ داﻟﺔ أﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ fاﻟﻤﻌﺮﻓﺔ ﺑــ :
f ( x) = 2 x + 1
-2ﺣﺪد داﻟﺔ أﺻﻠﻴﺔ ﻟﻜﻞ داﻟﺔ ﻣﻦ اﻟﺪوال اﻟﺘﺎﻟﻴﺔ : f ( x) = 2 -a
\ ∈ F ( x) = 2 x + C / C -b
f ( x) = x 1 2 x + C 2
-c
f ( x ) = x3 1 4 x + C 4
-d
= )F ( x
= )F ( x
*` ∈ / n
f ( x ) = xn
1 x n +1 + C n +1
= )F ( x
-e
}f ( x ) = x r ; r ∈ `* − {−1
-f
1 x r +1 + C r +1 f ( x) = x
= )F ( x
1
= x2 2 32 x + Cte 3 -g
)(2 x + Cte
3
)+ 1 4
= )F ( x 2
(x
)+ 1
1 u r +1 + C r +1
2
(x
= )f ( x 1 4
= )F ( x
اﻷﺻﻠﻴــﺔ
ur ⋅ u ' :
-2ﺧـﺎﺻﻴـــﺔ : ﻟﺘﻜﻦ fداﻟﺔ ﻋﺪدﻳﺔ. إذا آﺎﻧﺖ Fداﻟﺔ أﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ ﻣﺠﺎل Iﻓﺈن ﻣﺠﻤﻮﻋﺔ اﻟﺪاﻟﺔ اﻷﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ Iهــﻲ : F + λ \ ∈ .λ ﺣﻴﺚ ﺑـﺮهــــﺎن : ﻟﺘﻜﻦ Fداﻟﺔ أﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ Iو λﻋﺪد ﺣﻘﻴﻘﻲ.
+ λ)' = F ' = f
ﻟﺪﻳﻨﺎ :
(F
F + λهـﻲ أﻳﻀـﺎ داﻟﺔ أﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ . I إذن : وﻣﻨﻪ :ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ Iهــﻲ . F + λ -3ﺧـﺎﺻﻴـــﺔ : ﻟﺘﻜﻦ fداﻟﺔ ﻋﺪدﻳﺔ ﺗﻘﺒﻞ داﻟﺔ أﺻﻠﻴﺔ ﻋﻠﻰ . I ﻟﻴﻜﻦ x0ﻣﻦ Iو y0ﻋﻨﺼﺮ ﺣﻘﻴﻘﻲ \ ∈ . y0 ﺗﻮﺟﺪ داﻟﺔ أﺻﻠﻴﺔ وﺣﻴﺪة Fﻟﻠﺪاﻟﺔ fﻋﻠﻰ . I ﺣﻴﺚ :
F ( x0 ) = y0
أﻣـﺜـﻠـــﺔ : ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ fواﻟﺘﻲ ﺗﺤﻘﻖ اﻟﺸﺮط . F ( x0 ) = y0
f ( x) = x + 1
-1
1 2 x + x + C = 1 2 F ( 2) = 1
ﻟﺪﻳﻨﺎ : وﺑﻤﺎ أن :
= )F ( x
1 2 x + x + C = 1 2 2 + 2 + C = 1 C = −3
ﻓﺈن : وﻣﻨﻪ : 2 x + 1
-2
F ( 2) = 1
2
F ( 0) = 0
= )f ( x
ﻟﺪﻳﻨﺎ :
F ( x ) = 2 Arc tan x + C
وﺑﻤﺎ أن :
F ( 0) = 0
C = 0 = 2 Arc tan x
ﻓﺈن : إذن :
)F ( x
f ( x ) = cos 2 x
-3 ﻟﺪﻳﻨﺎ :
وﺑﻤﺎ أن : ﻓﺈن :
+ C
)(2 x
⎞ ⎛π F⎜ ⎟ = 0 ⎠⎝2 C = 0
⎞ ⎛π F⎜ ⎟ = 0 ⎠⎝2 1 = )F ( x sin 2
1 sin 2 x 2
= )F ( x
-4ﺧـﺎﺻﻴـــﺔ : إذا آﺎﻧﺖ Fداﻟﺔ أﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ . I و Gداﻟﺔ أﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ gﻋﻠﻰ . I ﻓﺈن :اﻟﺪاﻟﺔ F + Gداﻟﺔ أﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ f + gﻋﻠﻰ . I
-5ﺧـﺎﺻﻴـــﺔ : آﻞ داﻟﺔ ﻣﺘﺼﻠﺔ ﻋﻠﻰ ﻣﺠﺎل Iﺗﻘﺒﻞ داﻟﺔ أﺻﻠﻴﺔ . ﻣﻼﺣـﻈــﺔ وﺧـﺎﺻﻴـــﺔ :
إذا آﺎﻧﺖ Fو Gداﻟﺘﻴﻦ أﺻﻠﻴﺘﻴﻦ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ ، Iﻓﺈﻧﻪ ﻳﻮﺟﺪ ﻋﺪد ﺣﻘﻴﻘﻲ λ F − G = λ ﺣﻴﺚ : -6ﺟــﺪول اﻟــﺪوال اﻷﺻﻠﻴــﺔ اﻻﻋﺘﻴـﺎدﻳـــﺔ : اﻟﺪاﻟــﺔ f
اﻟﺪاﻟــﺔ ) Fاﻷﺻﻠﻴــﺔ (
1
x+C
x
1 2 x +C 2
xn
1 n +1 x +C n +1
`∈n
xr
1 r +1 x +C r +1
}r ∈ _ − {−1
' un ⋅u
1 n +1 u +C n +1
`∈n
' ur ⋅u
1 r +1 u +C r +1
}r ∈ _ − {−1
1 x +1
Arc tan x + C
cos x
sin x + C
sin x
− cos x + C
) cos ( ax + b
1 sin ( ax + b ) + C a
a≠0
) sin ( ax + b
−1 cos ( ax + b ) + C a
a≠0
2
1 cos 2 x
= ) 1 + tan 2 ( x
tan x + C
ﻣـﻼﺣﻈـــﺎت
\∈ C
+ kπ
π 2
≠x
: ﺗﻄﺒﻴـﻘـــــﺎت : ﻓﻲ اﻟﺤﺎﻻت اﻟﺘﺎﻟﻴﺔf ﺣﺪد داﻟﺔ أﺻﻠﻴﺔ ﻟﻠﺪاﻟﺔ x2 − 1 x2 + 1
f ( x) =
-1
=
x2 + 1 − 2 x2 + 1
=
−2 + 1 x + 1 2
F ( x ) = x − 2 Arc tan x + C f ( x) = x
: إذن
x2 + 1
3
-2
1 2 x 3 x2 + 1 2 1 1 2 3 = x + 1 (2 x) ( ) 2
f ( x) =
1 2
F ( x) =
1
( x2 + 1) 3
1 1 +1 3
1 3 × 3 4
(x
=
3 8
(x
+ 1) 3
F ( x) =
3 8
= F ( x) =
4
2
x2 + 1
3
+ 1)
(x
+ x + 3) 2
2 3
4
: إذن
x2 + x + 3 1
(2 x
-3
+ 1)
3
( x 2 + x + 3) 2 f ( x) =
sin 3 x cos5 x
F ( x ) = tan 3 x ⋅ = f ( x) =
: وﻣﻨﻪ
+ 1) 3
2
(2 x 2
+1
4
=
f ( x) =
: ﻟﺪﻳﻨﺎ
3
(x
2
+ 1) x 2
-4 1 cos 2 x
1 tan 4 x 4
: ﻟﺪﻳﻨﺎ
-5
2 1 2 f ( x) = ( x + 1) 3 2 x 2 5 1 3 F ( x) = x 2 + 1) 3 + C ( 2 5 5 3 = x 2 + 1) 3 + C ( 10
: إذن