Литвин І.І., Конопчук О.М., Желізняк Г.О. Вища математика
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161
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8. ^ _
163
() lim () ;
() lim () ;
() lim () lim () ;
b
b
aa
db
a
a
c
ac
fxdx fxdx
fxdx fxdx
g x dx f x dx f x dx
E
DE
f
of
of
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f
³³
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& (
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R7 ( ( , (,
! =b,
0
() lim () ;
bb
aa
f
xdx f xdx
H
H
o
³³
7
()
f
x
< ' ! = , (
[;]cab
9 ! ,
00
() lim () lim ()
bc b
aac
f
xdx f xdx f xdx
H
HH
H
oo
³³ ³
.
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)
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8. ^ _
164
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7 !) ) ' “ ” .
1: "'!
2
2
1
dx
x
f
³
.
?;’;!!: + ) ) )
f
. C(
!
21
2
22
222
11 111
lim lim lim ( ) lim
22
bb
b
bbb b
dx dx x dx x
b
xx
f
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165
8. ^ _
2: "'!
1
0
ln
x
dx
³
.
?;’;!!: + , '
lnyx
!
.
0
x
0
limln
x
x
o
f
. C( !
1
0
ln
x
dx
³
=
1
0
lim ln
x
dx
H
H
o
³
.
"'!
1
ln
x
dx
H
³
!:
11
1
1
0
1
lim ln ln ln 1 lnxdxxx xdxxxx
x
H
H
H
HH
HHH
o
³³
.
u = ln x du =
1
dx
x
dv = dx v = x
",
1
00 0 0
00
2
ln
lim ln lim( 1 ( ln )) 1 lim ln 1 lim
1
1
1 lim 1 lim 1
1
xdx
HH H H
H
HH
H
HHH HH
H
H
H
H
oo o o
oo
³
.
/ (
0
lim ln
H
HH
o
.
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1
0
ln 1xdx
³
.
166
8.10. &:
1. ! ! .
2. / ! .
3. 5 ! < ! ! ?
4. &9 ( ! 7 () -
! .
5. C ( '’< '?
6. &9 ( '! 9, ( .
7. &9 ( '!
' .
8. C ) ) '!)< ( ?
9. "'! '’< , ' 0X , '-
') &
2
= , ) = 2, ) 0X.
10. "'! '’< , ' 0X , '-
') &
2
= , &= 1, & = 4, ) 0Y.
11. A < 9() V = 2+4t
3
(V – /). "'!
9, 9 9 3 (.
12. R ' ' , 7' 6 , 7
1& < 1 ?
13. "'! ( 9) )
18 ) 6 .
14. "'! !
15. )
³
3
2
2
)
1
3( dx
x
x
; ')
³
2
1
ln
e
dx
x
x
;)
³
4
ln
xdxx
.
16. )
³
8
1
3
2
7
dx
xx
; ')
³
1
1
2
52xx
dx
;)
³
1
0
arcsin xdx
.
17. )
³
2
1
3
2
)1
4
3( dx
x
x
; ')
³
4/
6/
2
sin
S
S
xctgx
dx
;)
³
2/
0
cos
S
xdxx
.
18. )
³
8
0
3
2
)1
3
1
( dx
x
; ')
³
S
S
xdxx 7sin5sin
;)
³
1
0
xarctgxdx
.
19. )
³
4
1
2
)
8
2
9
2( dx
x
xx
; ')
³
2/
2/
7cos2cos
S
S
xdxx
;)
³
xdx
1
2
ln
.
8. ^ _
167
20. )
³
3
1
3
)
3
9124
(
e
dx
x
xx
; ')
³
2/
2/
4sinsin
S
S
xdxx
; )
³
1
0
dxx
.
21. )
³
2
1
2
)
21
( dx
x
xx
; ')
³
2/
0
cos1sin3
S
xdxx
; )
³
2ln
0
dxxe
x
.
22. )
³
4/
6/
2
)
sin
1
(cos
S
S
dx
x
x
; ')
³
S
S
2
2/3
sincos1 xdxx
; )
³
e
dxx
/1
ln
.
23. )
³
6/
3/
2
)
cos
1
(sin
S
S
dx
x
x
; ')
³
3/
2/
cos3
sin
S
S
dx
x
x
; )
³
S
0
sin xdxx
.
24. )
³
6/
6
42
sin4sin41
S
xx
dx
; ')
³
2/
0
sin2
cos
S
x
xdx
; )
³
1
0
arccos xdx
.
25. )
³
3
1
)21(2 dx
xx
; ')
³
3/
2/
cos1
sin
S
S
t
tdt
; )
³
S
2
0
arctg xdx
.
26. )
³
5
4
3
)4( dxx
; ')
³
6/
0
sin
cos
S
xdxe
x
; )
³
3
0
arctg xdxx
.
27. )
³
3
2
23
)1234( dxxxx
; ')
³
1
0
2
xdxe
x
; )
³
xdx
1
2
ln
.
28. )
³
0
1
3
)2( dxxx
; ')
³
3
3
2
0
2
3 dxxe
x
; )
³
2
1
5
ln
dx
x
x
.
29. )
³
4
1
)
1
( dx
x
x
; ')
³
4
1
dx
x
e
x
; )
³
4/
0
2
S
xdxxtg
.
8. ^ _
168
30. )
³
9
1
1
dx
x
x
; ')
³
3
3
1
3
dxxe
x
; )
³
1
0
2
)(arcsin dxx
.
31. )
³
2/
2/
)sin(cos
S
S
dxxx
; ')
³
2
1
ln
e
dx
x
; )
³
2
1
2
log xdxx
.
32. )
³
3/
6/
22
)
sin
1
cos
1
(
S
S
dx
xx
; ')
³
1
2
1
2
1
dx
x
e
x
; )
³
3
4
2
sin
S
S
dx
x
x
.
33. )
³
4/
4/
2
)sin
cos
1
(
S
S
dxx
x
; ')
³
0
1
3
2
41 x
dxx
; )
³
1
0
)1ln(
e
dxxx
.
34. )
³
2/3
2/1
2
1 x
dx
; ')
³
3ln
2ln
2
1
x
x
e
dxe
; )
³
e
xdxx
0
3
ln
.
35. )
³
3
0
2
1 x
dx
; ')
³
5,0
5,0
91
3
x
x
dx
; )
³
S
1
2
sin dx
x
x
.
36. )
³
4/
0
2cos
S
xdx
; ')
³
e
dx
x
x
1
5
lnsin
; )
³
S
0
3
sin xdxx
.
37. )
³
4/
0
4sin
S
xdx
; ')
³
3ln
3ln
xx
ee
dx
; )
³
2
0
2
cos
S
xdxe
x
.
38. )
³
1
1
2
)
1
( dx
x
e
x
; ')
³
5,0
0
sin
cos xdxe
x
; )
³
2
0
2
cos
S
xdxx
.
39. )
³
2
1
2
)
1
( dxx
x
e
x
; ')
³
4/
4/
2
cos
tg1
S
S
dx
x
x
; )
³
1
0
23
dxex
x
.
8. ^ _
169
"'! 7 , ' .
40. – & + 2 = 0; & = 0, = –1, = 2.
41. 2 – 3& + 6 = 0; & = 0, = 3.
42. – & + 3 = 0; + & = 0, & = 0.
43. – 2& + 4 = 0; +2& – 8 = 0; & =
2
, & = 0, = 0, = 3.
44. y = x
2
; y = 0; x = 0, x = 3.
45. y = 3x
2
; y = 0; x = –3, x = 2.
46. & =
2
+ 1; & = 0, = –1, = 2.
47. & = 0,5
2
+ 2; & = 0, = 1, = 3.
48. = –
3
1
2
+ 3; & = 0, = 0, = 3.
49. y
2
= x; y t 0, x = 0, x = 3.
50. & = –
2
+ 2x+ 8; & = 0.
51. y = –
9
2
x
2
+ x; y = 0.
52. y = –x
2
+ 6x – 6; y = 0, x = 2, x = 3.
53. & =
x
1
; & = 0, = 1, = 3.
54. & =
x
2
; & = 0, =2, = 4.
55. y = cos x; y = 0, x = 0, x =
2
S
.
56. y = tg x; y = 0, x = 0, x =
4
S
.
57. y
2
= 4x; x = 1, x = 9.
58. y
2
= 9x; x = 4.
59. y = sin x; y = 0, x = –
2
S
, x = S.
60. y = sin x; y = 0, x = 0, x = 2 S.
8. ^ _
170
61. y = x
2
; y = – 3.
62.y = x
2
; y = 2x + 8.
63. y = x
2
; y = x + 2 .
64. & = –
2
1
2
– 4x + 10; & = + 2.
"'! '’< , ' , '
65. y = x
2
; y
2
=x.
66. & =
10
4
x
; & = 0, =2;
67. y = sin x, y = 0, x < [0; S];
68. y = 3x – x
2
; y = 0;
69. y = x
2
, y = 0, x = 2;
70. y = 10 – x
2
, y = 0;
71. y = x
3
, y = 0, x = 1;
72. y = arcsinx, y = 0, x = 1;
73. y = 2x – x
2
, y = 0;
74. >( ! v = (24t – 6t
2
) /. C(: ) 9, ( !)
3 ( ! ; ') 9, ( !) ) (; ) 9, -
( !) ( ! ( .
75. ! (! (< ! (
. /9 < 9() v = (4t +5) /, ( – 9()
v = (6t
2
+2t) /. & ( ( ( ( '( ! 5 ?
76. A, C 9() v = (39,2 – 9,8t) /.
C '9 ( .
77. / < 0,02 ( (<) 60 &. R ' <
, )! 0,12
2
.
78. / < ( 0,1 . H 20 & <
0,01 . R ' ' , 7' ( 0,12 ( 0,14 ?
79. +(! <). "'! '!
) , 7 =0,08 ( R = 0,04 . %
900
/
3
.
80. "'! ( ( , 0,80,50,3.
@ ( ().
8. ^ _