Oloklirwmata
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ΦΡΟΝΤΙΣΤΗΡΙΑ ΗΛΙΑΣΚΟΣ ΥΠΗΡΕΣΙΕΣ ΠΑΙΔΕΙΑΣ ΥΨΗΛΟΥ ΕΠΙΠΕΔΟΥ Θετικής - Τεχνολογικής Κατεύθυνσης Μαθηματικά Γ΄Λυκείου Ολοκληρώματα ΣΤΕΦΑΝΟΣ ΗΛΙΑΣΚΟΣ e-mail: [email protected] www.iliaskos.gr
description
Ολοκληρωτικός Λογισμός
Transcript of Oloklirwmata
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e-mail: [email protected] www.iliaskos.gr
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F x f x=l] ]g g[ ! D
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G x F x c= +] ]g g F R! I
*IG x F x c= +] ]g gF R!
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I f x dx] g# f x dx F x c= +] ]g g# F R!
I
f x cdx f x +=l] ]g g# F R!
dx c0 =# dx x c1 = +#
xdx x c2
= +# f x f x cf x dx 2
= +l] ] ]g g g# ,x dx x 1 1
1
!a
a= + -a
a+
# ,f x f x dx f x c1 11
!a
a= + + -a
a+
l] ] ]g g g# ,lnx dx x c x
1 0>= +# lnf xf x dx f x c= +l]] ]gg g#
-
e dx e c = +# e f x dx e cf x f x= +l]] ]gg g# lndx c
aaa= +# lnf x dx c
f xf x
aa
a= +l]] ]gg g# xdx x chn vyo=- +# f x f xf x dx chn vyo=- +l]^ ] ]^gh g gh# xdx x cvyo hn= +# f x f xf x dx cvyo hn= +l]^ ] ]^gh g gh# x x c
1hn
v{=- +# f xf x dx f x chn v{=- +l ]]^ ]^ghg gh#
x x c1
vyof{= +# f x
f x dx f x cvyo f{= +l ]]^ ]^ghg gh#
,f x dx f x dx R!m m m=] ]g g# # f x dx f xg x dx g x dx=+ +] ]^ ] ]g gh g g# # # , ,f x g x dx f x dx g x dx R!m n m n m n+ = +] ]^ ] ]g gh g g###
L
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x dx x dxx c1 1
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LL
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l l] ] ] ] ] ]g g g g g g# #
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f x g x dx f x g x f x g x dx= -l l] ] ] ] ] ]g g g g g g# #
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x e dx e x dx2 1 4 2 1x
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I e xdx e xdx hn hn= = =l] g##e x e x e x e xdxdx hn hn hn vyo- - ==l^ h# #e x e xdx hn vyo- =l] g#e x e x e x dx e x e x I +hn vyo hn hn vyo- + - = - -^ h#2I e x e x I e x e x c2
+hn vyohn vyo= - = - +
-
LLL lnx
lnx
ln ln lnxdx x xdx x x x x dx1= = - =l ###
ln lnx x dx x x x c1- = - +# ln lnx dx x x dx1 1 1+ = + + =l] ] ]g g g# #
lnx x x x dx1 1 1 11+ + - + + =] ] ]g g g#
ln lnx x dx x x x c1 1 1 1 1+ + - = + + - +] ] ] ]g g g g#
P x] g xQ] g x
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x dx Q xU x dxP +] ]]g gg# # [P] g
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LLL Q x] g x xa b c+ +^ h 4 0
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t t= - = - +#
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x xx
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x A x A x3 2 11 2- = - + -] ]g gx 2= A A A1 0 1 11 2 2+- = + =-] g x 1= A A A2 1 0 21 2 1+- = - + =] g x x
xx x1 2
31
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x A x x x x3 1 2 2 2 11 2 +- = - - + - - -] ] ] ]g g g g
-
x A x A x A3 1 4 3 2 51 2 1 1- = - + - + -] ]g g [
A A1 0 1 +- = =
13 1
xx
x x x x2 1 12
21
--
- = - + - + --] ] ]g g g
I x xx dx x x x dx1 2
31
112
21
+= - -- = - + - - -] ] ]cg g g m# #
ln lnI x x x c1 12 2= - - - - - +
LLL I x xAx B dxa b c= + +
+# 4 0
-
x xx x
x xx
1 12 1
11
12
- +- - = -
- + +] ]g g
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x dx1 12 1
11
12
+= - +- - = -
- + +] ] bg g l# #ln lnI x x c1 1=- - - + +] g
LY I xAx B dxa b c= +
++ l^ h# 4 0
-
W
W
LL t h x= ] gh x] gx h t dx h t dt1 1&= =- - l] ]^g gh I f h t h t dt1 1= - - l]^ ]^gh gh# t h x= ] g[
L
1 ,x x x x2
22
1 2 hn vyo vyo vyo= - = +
I xdxhn= # 1 2I xdx x dx x x c2
22 4
hnvyo hn= = - = - +##
x x 1 hn vyo+ =
[hn u x du xdx&vyo hn= =-
[vyo u x du xdx&hn vyo= =
I xdxhn= # I xdx x xdx3 2hn hn hn= =# #u x du xdx&vyo hn= =-
1u x u x u x x u1 + + +vyo vyo hn hn= = = - = -
I u du u u c1 32
3
+= - - =- + +] g#I x x c3
vyovyo=- + +
I x xdx3 2hn vyo= # I x xdx x x xdx3 2 2 2hn vyo hn hn vyo= =# # u x du xdx&vyo hn= =-
1u x u x u x x u1 + + +vyo vyo hn hn= = = - = -
-
I u u du u u du u u c1 3 52 2 2 4
3 5
+= - - = - + =- + +] ]g g# #I x x c3 5
3 5vyo vyo=- + +
LL
xa b+ [
dx
I x x dx1= +# u x u x x u dx udu1 1 1 2 + + &= + = + = - =
I u udu u u du u u cu1 2 2 2 2 5 33
2 4 25
+= - = - = - +] ]g g##1I x x c2 5
13
5 3
= + - + +^ ^h h
LLL
H
lnu e x u dx u du1 + &= = =
I ee dx1
1
= +-#
lnu e x u dx u du1 + &= = =
I uu
u du11 1= +
-#
u uu
uA
uB
11
1+- = + +] g
u A u Bu1 1- = + +] gx 0= A B A1 0 1+- = + =-
x 1=- A B B2 0 1 2+- = + - =] g u u
uu u1
1 11
2+- = - + +] g
I u uu du u u du1
1 11
2= +- = - + + =] bg l##
2ln ln ln lnu u c I e e c1 2 1 +- + + + =- + + +
LY
lnx
lnu x x e dx e du + &= = =
-
ln lnI x x dx3= + -] g# lnu x x e dx e du + &= = =
I u u e du3 u2= + - =] g#u u e u e du3 2 1u u2+ - - + =] ]g g#
u u e u e due3 2 1 2u uu2+ - - + =+] ]g g #u u e u e e c3 2 1 2u u u2 ++ - - + + +] ]g g
ln ln lnI x x e x e e c3 2 1 2ln ln lnx x x2= + - - + + +] ]g g Y
x a -
x u dx u&ahn avyo= =
x a-
x u dx uu
&hna
ahnvyo= =-
x a +
x u dx u du1
&af{ a vyo= =
v
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-
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a
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a
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a
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a
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a
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a
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u x du dx&=- =-
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a
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a a] ]g g# #If x f x- =] ]g gI f u f x f xdu dx dx2
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a a a] ] ]g g g# # # I
I f x dx=a
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a
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u x du dx&=- =-
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a
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a a] ]g g# # LL
,I I1 2
,f xhn^ h,f xhn^ h piu x du dx&= - =-
, f xvyo] g , f xvyo] g piu x du dx2 &= - =-
,f xhn^ h,f xvyo] g piu x du dx2 &= - =-
,f x] g,f xl -] gu x du dx&l= - =-
-
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pi pi
0
2
0
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piu x du dx2 &= - =-
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pi0
2
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0
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LLL f x1- ] g
f x] g u f x dx f u dux f u1 &+= ==- l] ] ]g g g
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a
b
a
b- ] ]g g# #
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f f 1+a b b a= =-] ^g h f f 1+b a a b= =-^ ]h gu f x x f u dx f u du1 + &= = =- l] ] ]g g g x u f 1&a a b= = =- ] g x u f 1&b b a= = =- ^ h f x uf u uf udx du f u du1 = = - =
a
b
b
a
b
a
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f f u du f u duf f u dua a b b ab ba =- + = - +a
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a
b
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a
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