Ασκήσεις εφ' όλης της ύλης στα ολοκληρώματα, από το ανανεωμένο βιβλίο του Γιώργου Μαυρίδη για την Γ΄ Λυκείου. Οι ασκήσεις είναι μια φιλική προσφορά του φίλου και συναδέλφου, Γιώργου Μαυρίδη.

1. 9 786188 021495
ISBN: 978-618-80214-9-5

2. .»

3. “
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4. 2013

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7. 486
102
f :
f x 0 x
x u
0
1
f t dt du 2x 2 x .
i) g :
x u
0
1
g x f t dt du 2x 2 x
g
g .
ii) N
1
0
f t dt 2.
iii) N
f x y
x 1.
i)
g x 0 x .
,
1 u
01
g 1 f t dt du 2 2 0.
,
g x g 1 x .
g 0x 1 g 1 0.

8. 487
) f
u
0
f t dt
x u
01
f t dt du . , g
x u
01
g x f t dt du 2x 2
x u
01
f t dt du 2x 2
x
0
f t dt 2 x .
ii) i) g
0x 1, ,
Fermat
g 1 0
,
1
0
f t dt 2 0
1
0
f t dt 2.
iii)
f x 0 x .
1 1
0 0
E f x dx f t dt 2

9. 488
103
f 0 f 2
x 2
0
1
f x f 2 f t dt
3
i) f 2 0 .
ii) N
f x 0
iii) f .
iv)
2
x x x 1 x 2.
i) f 2 0.
x
0
2
0
1 1
f 0 f 2 f t dt 0 0 0
3 3
f 0 f 2
f 2 0.
ii)
2
2
0
1
f 2 f 2 f t dt
3
2
2
0
2
f 2 f t dt
3
2
2
0
3
f 2 f t dt
2
(1)
x
2
0
f t dt 0 x 0, 2
x
2
0
1
f x f 2 f t dt
3
f 2 0.

10. 489
2
f t
0, 2
2
f 2 0 .
2
2
0
f t dt 0
(1) f 2 0 .
x
2
0
f t dt 0 x 0, 2 .
x
2
0
1
f x f 2 f t dt 0
3
x 0, 2 .
iii)
2
f t
x
2
0
f t dt
f
x
2
0
x
2 2
0
2
1
f x f 2 f t dt
3
1
f 2 f t dt 0 f x
3
f x 0
2
f x
1
f x
x 0, 2
1
x
f x
x 0, 2
1
x c
f x
x 0, 2 c
1
f x
x c
x 0, 2 .

11. 490
x o
0
2
0
1
f 0 f 2 f t dt
3
1 1 1 1
f 0 f 2
3 0 c 3 2 c
3 2 c c 6 3c c
2c 6 c 3.
1
f x
x 3
x 0, 2 .
iv) H
2
g t f t , t 0, 2
2 2
2
1 1
22 22
2 2
11 1
2 2
11
E g t dt f t dt
3 t1 1
dt dt dt
t 3 3 t 3 t
1 1 1 1 1 1
dt 1
3 t 3 t 3 2 3 1 2 2
c.
x
1
f 0 f 2 .
3

12. 491
104
x
0
t
f x dt 1
g t
x
x
0
t
g x dt 1
f t
x .
i) N
f x 0 x
g x 0 x .
ii)
f x g x x .
iii) f.
iv)
x
1
1
f t
x
x x y y x 1.
i)
f x 0 x
g x 0 x .
t
g t
t
f t
f g

13. 492
f g
.
0
0
t
f 0 dt 1 0 1 1 0
g t
0
0
t
g 0 dt 1 0 1 1 0
f t
f x 0 x
g x 0 x .
ii)
t
g t
t
f t
.
f x g x
x
0
t x
f x dt 1
g t g x
x
x
0
t x
g x dt 1
f t f x
x .
f x x
f x f x g x
x
g x x
g x f x g x
x .
,
f x g x
f x g x
x .
lnf x lng x x
lnf x lng x c x c .

14. 493
x
lnf 0 lng 0 c ln1 ln1 c c 0.
lnf x lng x x
f x g x x .
iii)
x x
f x
g x f x
x .
,
2f x f x 2x x
2 2
f x x x
2 2
f x x c x c .
x
2 2
f 0 0 c c 1.
2 2
f x x 1 x .
f x 0 x ,
2
f x x 1 x .

15. 494
iv)
1
f t
.
x
1
1
h x dt
f t
2
1 1
h x 0
f x x 1
x .
h
1
1
1
h 1 dt 0
f t
,
h x 0 h x h 1 x 1
h x 0 h x h 1 x 1.
1 1 1
0 00
E h x dx h x dx x h x dx
11
1
0 20 0
1
xh x x h x dx 1h 1 0h 0 x dx
x 1
1 1 1
2 2
2 000
2x
0 0 dx x 1 dx x 1
2 x 1
2 2
1 1 0 1 2 1

16. 495
105
f :
1f x f x
t
2x 1
2
1
e
dt e e
t
x .
i) N
f x 0 x .
ii)
2
f x x 1 x .
iii) A F f
F
x x x 0 x 1.
i)
1
t
2
e
t
*
1f x
t
2
1
e
dt
t
x f x
*. x
f x 0, .
x .
f x 0 x .

17. 496
ii)
2
f x f x1
t x 1
2
1
1
e dt e e
t
x
2
f x f x1
t x 1
1
1
e dt e e
t
x
2
f x f x1
t x 1
1
e dt e e x
2
f x f x1
t x 1
1
e e e x
2
f x1 1
f x 1 x 1
e e e e x
2
f x1
f x x 1
e e x
2
f x1
f x x 1
x
2 2
f x x 1 x
f x 0 x
2
f x x 1 x .
iii) F .
1
0
E F x dx .
2
F x f x x 1 0 x .
F
F x F 1 0 .

18. 497
F x 0 x 0,1 .
1 1
0 0
E F x dx F x dx
1 1
1
0
0 0
x F x dx xF x x F x dx
1
0
1 F 1 0 F 0 xf x dx
1
2
0
x x 1dx .
2
u x 1 .
2 2
u x 1
2 2
u du x 1 dx 2udu 2xdx udu xdx.
x 0 u 1
x 1 u 2.
1 1
2 2
0 0
x x 1dx x 1xdx
2
1
uudu
2
2
1
u du
3
23 3
1
2u 1
3 3 3
2 2 1
.
3
2 2 1
E
3

19. 498
6
f : :
f 1 f 1
f x 0 x *
x
1
f x
1 2 f t dt
x
x *.
i) N f 0 f.
ii)
2
x 1
f x xe x .
iii) N fC y x.
iv) N f
f 1
f .
i) A
x
1
f x
1 2 f t dt
x
x * (1)
x
1
f x x 2x f t dt x *. (2)
f t ,
x
1
f t dt
O (2)
x 0
x 0 x 0 x 0 x 0 1 1
limf x limx lim2x lim f t dt 0 0 f t dt 0
f
x 0
f 0 limf x 0. H f
*. f
, 0 0, . ,
(2)
1
1
f 1 1 2 f t dt 1 0 1 0
f 1 f 1 1 0 .

20. 499
f x 0 x , 0
f x 0 x 0, .
ii) A (2)
x
1
f t dt
f
x x x
1 1 1
f x x 2x f t dt 1 2x f t dt 2x f t dt
x
1
f x
1 2 f t dt 2xf x 2xf x
x
1
1
2x f x
x
.
f x 1
2x
f x x
2
ln f x ln x x (3)
, 0 x 0 f x 0 (3)
2
ln f x ln x x x 0
2
1ln f x ln x x c x 0 1c
x 1
1 1 1ln f 1 ln1 1 c ln1 ln1 1 c c 1.
,
2
2
x 1
2x 1
2x 1
2x 1
ln f x ln x x 1
ln f x ln x lne
ln f x ln xe
f x xe
f x xe

21. 500
0, x 0 f x 0 (3)
2
lnf x ln x x
2
2lnf x ln x x c x 0 2c .
x 1
2 2 2lnf 1 ln1 1 c ln1 ln1 1 c c 1.
2
2
2x 1
x 1
2x 1
lnf x ln x x 1
lnf x ln x lne
lnf x ln xe
f x xe
x 0 f 0 0 .
2x 1
f x xe x .
iii) f x x
2 2x 1 x 1
2x 1
2
xe x 0 x e 1 0
x 0
x 0
x 0
fC y x A 1, 1 ,
O 0, 0 B 1,1 .
iv)
2 2 2x 1 x 1 x 1 2
2 2x 1 2 x 1
f x xe x e x e x 1
e 2x e 0
f 1 1
fC 1f
C y x
1
2

22. 501
1 fC y x ,
1
1 1
E
2x 1
f x x x e 1
1, 1
0 1
1
1 0
E
0 1
1 0
0 10 12 2x 1 x 1
1 001
f x x dx x f x dx
xe dx xdx xdx xe dx
2
0 10 12 2
2x 1 x 1
1 01 0
1 x x 1
e e
2 2 2 2
1 1 1 1 1 1
0 0
2e 2 2 2 2 2e
1
e
1
2
e
x 1 0 1
x
2x 1
e 1
f x x
0
00
0 00

23. 502
107
z
z 2
i) z
f
f ;
ii) N
2
2
f x dx.
i) z x yi M x, y
z 2 Im z 0 .
2 2
2 2
2 2
2 2
2
x y 2
x y 4
y 4 x
y 4 x
y 4 x
z
2
f x 4 x , x 2, 2 .

24. 503
ii)
2
f x 4 x 0 x 2, 2 .
E
fC x x.
z 2 Im z 0
z fC
c
O 0, 0
fC
c
2 21 1
E
2 2
2
2
f x dx 2
z
fC
O x
y
fC
22