ΟΛΚΛΗΡΩΜΑΤΑ.pdf
10
3 Maθηματικά Γ΄ Λυκείου Γ. Ολοκληρωτικός Λογισμός
Transcript of ΟΛΚΛΗΡΩΜΑΤΑ.pdf
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3Ma
.
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- 1 -
1. :
3
1 2 23
6
1234I dxx(1 x)
2 3
2 40
xI dx1 x
1 x
3 20
xeI dx1 x
3
40
xI dx1 x
e
51
ln xI dxx
3
61
ln xI dxx 1
2e
71
I ln x 1 dx 3e
8e
I ln x x dx
2
9I f x dx
xx x, x 0
f xxe , x>0
3
10
6
I x ln 1 x dx
0x 1
111
e dx
2e
12e
2ln x ln xI dxx ln x
2
2
4
13I x dx
2e
141
I ln x dx
e
215
1
I x ln xdx x
16 2 2 x
2
2x 2 x x eI dxx x e
3
17 2
6
3I dxx x
1x
180
I x 3 dx 3
2
e
19e
dxIx ln x ln(ln x)
2. :
) 0 0
x f x dx f x dx2
)
2 2
0 0
x f x dx x f x dx
.
3. e e
2 2
1 1
ln x dx ln xdx .
4. :
) 2 2x
t x
1
1e dt e e2
) x
3 3e x 1
t t
0 0
e dt e dt,
x .
5. :
) x
2
1x
f x tdt ) x
0
g x x t dt .
6. ) 1
t
I t x ln xdx , t 0 .
) t 0limI t
.
) 1
1
I f x dx
x , x 0
f xxlnx, x>0
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- 2 -
7. f f x 0 x
t 2 x 1
1
e f (t)dt x 1 3e 3 , x . ) f 1 .
) x
t
1
g x f t e dt .
8. :x
2
0
2 t dt 0,x [0, ]
.
9. f 2
2x
t
x
f x e dt
. f x 0
3,2
.
10. f 1
0
f x dx e . 0,1
f e 2 .
11. f , , 0 , , f (t)dt 0
. x1g x 2 f (t)dt,
x 0x ,
0x ( , ) :) g 0 0x ,f (x )
'x x . ) 0 0g(x ) 2 f (x ) .
12. f . g x 1
x
g(x) f (t)dt
.
13. f f x 2 . 2x 5x
2
0
g x x 5x 1 f (t)dt
, :
) g 3 g 0 0 .) g x 0 3,0 .
14. :
) x
3
F x ln t 2t dt ) x
2
3
G x t 4dt
) 2x 4x
2
tK x dtt
) 2 x
x 3
ln t 2L x dt
4 t
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15. x
2
1
f(x) t 1dt xg(x) e x , x . ) f . ) g . ) fog ,
.
16. f , :x
t x xe f (t)dt e e e f (x),
x, R .
17. f 0,3
,
3
0
xf x dx f x 2 x
.
18. f : , , 3f .
4 44 f x dx
, 3f x x .
19. f ln x t
t0
tef (x) dt1 e
, 0x .
) f .) f 0,1 .
) :lnx
t0
1 tf dt,x 0x 1 e
.
20. A) f , , f R.
i. f ( )
1
a f ( )
xf (x)dx f (x)dx
ii. :f ( )
1
f ( )
f (x)dx f (x)dx f ( ) f ( )
) xf x e x, x R . 1-1 1 e
1
1
f (x)dx
.
21. xf x e 3x 4 , x .) f .
) e 1
1
3
I f x dx
.
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- 4 -
22. f , g , :x 1
2
1 x
f (t)dt g(t)dt x 2x 1 ,
x f x 0 1 2, 1 21 . ) N g x 0 1 2, . ) 0 1 2x , 0g (x ) 2 . ) g , f f
, 0x .
23. , , :
)
x
02x 0
t tdtlim
x x
)
1x 1x
2x0
elim dtln e t
)
x
22x 1
ln t dtt
limx 1
)
2xt
x 12
1lim t e dtx 1
)
x3t 5
1xx
2t 3
0
e dtlim
e dt
)
2
2x
t
x 23x 2
lim 2t t e dt
24. ) x
t 1
a
e dt,x 1 . x
, .
) [2,4], 1 t 1e dt
.
25. x 1
3
x
F x ln t dt.
.
26. f R f x 0 x . :
1
0
f(x) 1dxf(x) f(1 x) 2
.
27. f : , f (x) 0 . F
F x f(x t)dt
, , , 0x R
0F (x ) 0 F(x)=0 x.
28. f f (x) 0 x 1
2 2
0
f x 1 2x tf (xt)dt , 21g x x ,x
f(x) .
: i. 2f (x) 2xf (x). ii. g .
iii. 21f(x) ,x .
1 x
iv. xlim (xf (x)2x).
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29. f [,] f x f x c x [,] , c .
: a
f (x)dx ( ) f (f ( ) f ( ))2 2
.
30. f, :1
1 x x
0
e f(x)dx f(x) e ,x R
31. ) f , . :
i. f , f(x)dx 0.
ii. f , 0
f(x)dx 2 f(x)dx.
)
2
x dxx 1
.
32. . 2
1f(x) ,x 0x 1
. f
x 1
2xx
1lim dtt 1
. x 1g(x) ,x 1ln x
. g
2x 1
xx 1
t 1lim dtln t
33. , f :x
t x
0
f (x) 1 e f (t)dt x.
34. 3f x x 4x , x x x 1 x 3 .
35. f x 3x 6 ln x x x .
36. 2f x x 4 , g x 5x x 0 x 2 .
37. 2f x x 3x 2 3 2g x x x 2x 4 .
38. 1,1 1 2, 2f x 1 x . 1 2, fC .
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39. 2f (x) x , x[0,1]. , () x x , f , () , y y x 1 .
40. ) h, g [,] h x g x
x , , h(x)dx g(x)dx.
) R f : f(x)f(x) e x 1
f 0 0. :i. f f .
ii. x f(x) x f (x)2
x 0 .
iii. f ,
x 0 x 1 x x : 1 1 f(1)4 2
.
41. f(x) x ln x,x 0 ,f 0 1.
) () f xx x 1 , x
) 0
lim ( ) .
) 1 sec, ()
1 .5
42. f [0, ) :i. f .ii. f 0 0, f 1 1. iii. 0 f ,
y y y f f , xx x . f .
43. f R f(0)=21
xe (f(x) f (x)) x f (x)
) xxf(x) ,x
1 e
.
) i. f x f x x x .
ii. 2
x
2
x dx1 e
) i. 2
0
0 f(x)dx4
ii.
0
2
4 f(x)dx 14
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44. f [0, ) [0, ) : f (x) xe f (x) e x 0 .
) f f(0) = 0.
) f (x) x x 0 .
) x
2
1
F(x) x f (t)dt x (0,1) ,
CF , 2A 1 ,3
.
) f 1 xf (x) ln(x e ) ,
x 1
1
xx
lim f (t)dt
.
45. f,g :
2 2
2
f x g x 1
f x g x 0 x g 0 1 .
) i. g x f x g x , x .ii. g ( ,0] [0, )
1. ) i. f .
ii. f 0.
iii. f ,
(): y x x 1 1 lng 12 .
46. 5 3f x x x x,x . ) 1-1.) xx R, f(e ) f(1 x) .) f 0,0 f 1f .) 1f ,
xx x 3 .
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47. f f 1 f 3 1 3f x f 3x , x .) 1 3
.
) , 3
0
f (x)dx 3( 3 1) :
i. 1
3
0
f (x)dx 3 1
ii. 1
x33
0 0
x x 3 xf (xt)dt f (t)dt :1. N .2. N 0,3 2 .3. x 2 x .
48. f , 2f x x 2x g , 2g x x 8x . x , .
49. f R f(1)=1 3x
1
1g(x) zf(t)dt 3 z (x 1) 0z
x , *z i .
) g R g (x) .
) z
zz 1 .
) 2 1Re z 2 . ) f 2 , f 3 0 2,3 f 0 .
50. f (0, ) :
x
1 x 1fx e
1 f (1)e
.
) : 1xf (x) x e ,x 0
) i. f x x 1 .
ii. 2
1
2f (x)dxe
.
) 3f (x)g(x)x
, E(t) gC x x
x 1 x t t 1 .
) tlim E(t)
.
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51. f 0 x x
1 dt f(t)
x1 1 f(x) .
) f(1) .
) 2 f(e) , e
1 dx f(x) .
) f 0 x .) f .
) : e
1 2
e
1 1dx
xlnf(x) I dx
xf(x) I .
)
x
1 1 x dt f(t)
x 11lim .
52. 2x
1
1f (x) dt1 t
, 0x . 11
4)5(f)7(f154
.
53. f : x2 t
0f (x) x e f (x t)dt
x .
54. ) x > 1 2x
x
x dt xln2x lnt lnx
.
) 2x
xx +
dtlimlnt .
