|
8.211.
8.212.
C
x
arctg
x
x
x
2
2
3
ln
2
2
2
8.213.
8.214.
C
x
x
2
/
3
1
1
8
3
8.215.
C
x
x
x
5
25
5
25
ln
5
25
2
8.216.
x
x
t
C
t
arctg
t
t
arctgt
1
,
3
1
2
3
2
1
ln
2
2
C
x
x
16
8
cos
4
2
cos
C
x
4
sin
3
8
3
1
,
1
1
ln
2
3
1
2
2
x
x
t
C
t
t
t
t
t
x
x
t
C
t
t
t
3
,
1
1
ln
2
8.217.
C
x
x
x
x
1
ln
24
24
6
4
12
12
6
4
8.218.
1
1
,
1
1
ln
1
2
2
x
x
t
C
t
t
t
t
8.219.
C
x
arctg
x
x
x
6
6
6
5
6
6
2
5
6
8.220.
C
x
x
arctg
3
2
6
2
3
6
8.221.
C
x
x
x
1
2
ln
6
2
6
2
3
6
6
3
8.222.
4
3
5
1
,
3
8
5
8
x
z
C
z
z
8.223.
2
/
1
3
/
2
3
1
3
x
t
C
t
t
8.224.
C
x
x
x
x
x
3
3
5
6
5
2
3
2
5
18
11
72
4
8.225.
C
x
x
x
3
4
16
5
4
8
7
4
2
/
3
2
2
/
5
2
2
/
7
2
8.226.
x
x
z
C
z
arctg
z
z
z
3
2
2
1
3
1
2
3
1
1
1
ln
6
1
8.227.
x
x
t
C
t
t
t
arctg
t
t
3
2
2
1
1
ln
3
1
3
1
2
3
1
1
ln
6
1
8.228.
C
x
x
x
x
2
/
1
2
2
/
3
2
2
/
5
2
2
/
7
2
1
1
1
5
3
1
7
1
8.229
C
x
x
x
x
x
3
2
2
2
3
1
1
1
8.230.
C
x
arctg
x
x
4
3
4
3
4
3
1
3
2
1
1
1
1
ln
3
1
Aniq integralga keltiriluvchi masalalar. Aniq
integralning ta’rifi va uning asosiy xossalari. Nyuton-
Leybnits formulasi. Aniq integralda o’zgaruvchini
almashtirish. Bo’laklab integrallash.
ANIQ INTEGRAL
§ 9.1. Aniq integralni hisoblash
1.
[a,b] kesmada
x
f
funksiya aniqlangan bo’lsin. [
a
,b] oraliqni
b
x
x
x
a
n
...
1
0
nuqtalar bilan n ta
bo’laklarga ajrataylik. Har bir [
i
i
x
x
,
1
] kesmadan bittadan
i
i
i
i
x
x
1
nuqta olib,
n
i
i
i
x
f
J
1
yig’indi
tuzamiz, bunda
1
i
i
i
x
x
x
.
J
-ko’rinishdagi yig’indi,integral yig’indi deyiladi. Uning
0
max
i
x
dagi
limiti, (u mavjud va chekli bo’lsa)
x
f
funksiyaning
a
dan
b
gacha aniq integrali deyiladi hamda
b
a
n
i
i
i
x
x
f
dx
x
f
i
1
0
max
lim
(9.1)
ko’rinishida yoziladi.
2
.Aniq integralning xossalari.
1)
const
dx
x
f
dx
x
f
b
a
b
a
;
; (9.2)
2)
b
a
b
a
b
a
dx
x
g
dx
x
f
dx
x
g
x
f
; (9.3)
3)
a
b
b
a
dx
x
f
dx
x
f
; (9.4)
4)
0
a
a
dx
x
f
; (9.5)
5)
b
c
c
a
b
a
dx
x
f
dx
x
f
dx
x
f
; (9.6)
6) Agar
x
f
y
funksiya [
a
,b] kesmada uzluksiz bo’lsa, u holda
b
a
,
topiladiki,
a
b
f
dx
x
f
b
a
; (9.8)
bo’ladi.
8) Agar
x
f
y
juft funksiya bo’lsa, u holda
a
a
a
dx
x
f
dx
x
f
0
2
; (9.8)
8) Agar
x
f
y
toq funksiya bo’lsa, u holda
0
a
a
dx
x
f
(9.9)
3.
Aniq integral Nyuton-Leybnits
a
F
b
F
x
F
dx
x
f
b
a
b
a
(9.10)
formulasi orqali hisoblanadi.
4
.
b
a
dx
x
f
integralni hisoblash uchun
t
t
x
,
almashtirishni qo’llaymiz. Agar [
;
] kesmada
t
f
t
t
x
,
,
funksiyalar uzluksiz va
b
a
,
bo’lsa, quyidagi
dt
t
t
f
dx
x
f
b
a
(9.11)
tenglik o’rinli.
5.
[a,b] kesmada
x
x
u
u
,
funksiyalar uzluksiz hosilalarga ega bo’lsa, quyidagi bo’laklab integrallash
formulasi o’rinli bo’ladi:
b
a
b
a
b
a
du
u
ud
(9.12)
“
A”guruh
Integrallarni hisoblang (Nyuton – Leybnis formulasini qo’llash yo’li bilan).
9.1
а
)
1
0
2
1
x
xdx
в
)
dx
x
1
0
1
9.2
1
2
2
5
11
x
dx
9.3
13
2
5
4
3
x
dx
9.4
3
0
1
1
dy
y
y
9.5
dt
T
t
T
2
/
0
0
2
sin
9.6
16
0
9
x
x
dx
9.8
dx
e
e
x
x
1
0
4
1
9.8
2
0
0
26
3
a
b
x
dx
9.9
1
0
2
2
1
x
xdx
9.10
e
x
x
dx
1
2
ln
1
9.11
e
dx
x
x
1
lg
1
9.12
2
1
2
/
1
x
dx
e
x
9.13
n
a
n
n
x
a
dx
x
2
/
0
2
2
1
9.14
3
1
ln
1
e
x
x
dx
9.15
2
/
3
2
/
1
4
4
3
8
5
8
5
x
x
dx
x
9.16
2
/
0
2
a
a
x
a
x
adx
9.18
3
2
2
2
3
2
x
x
dx
9.18
1
0
2
5
4
x
x
dx
9.19
2
1
3
x
x
dx
9.20
1
5
,
0
2
2
8
x
x
dx
“B”guruh
9.21
2
/
2
/
cos
1
x
dx
9.22
2
/
0
5
2
sin
cos
xdx
x
9.23
dx
x
x
2
/
2
/
3
cos
cos
9.24
/
0
0
2
sin
dx
t
9.25
4
/
2
/
3
3
sin
cos
x
xdx
9.26
4
/
0
4
d
ctg
9.28
/
2
/
1
2
1
sin
dx
x
x
9.28
2
/
2
/
4
2
sin
cos
dt
t
t
9.29
8
/
12
/
2
2
cos
x
dx
9.30
dx
x
0
1
3
2
9.31
3
1
3
4
x
dx
9.32
1
0
2
1
1
5
dx
x
9.33
2
/
3
0
2
2
3
4
1
3
dx
x
x
9.34
dx
e
x
2
2
4
3
9.35
6
/
2
/
4
cos
2
1
3
sin
dx
x
x
9.36
2
0
2
4
2
x
dx
9.38
2
3
2
2
9
x
dx
9.38
6
5
2
4
3
x
x
dx
9.39
2
/
2
/
cos
1
sin
x
xdx
9.40
1
0
2
8
3
x
xdx
9.41
4
0
4
1
dx
e
x
9.42
4
1
1
dx
x
x
9.43
7
1
4
3
x
dx
9.44
1
1
3
1
dx
x
9.45
5
1
2
3
x
dx
9.46
1
0
3
1
2
z
dz
9.48
0
2
3
cos
2
cos
dx
x
x
9.48
5
0
3
1
x
xdx
9.49
3
ln
2
ln
x
x
e
e
dx
9.50
2
/
0
cos
2
d
9.51
1
0
2
2
x
y
dx
x
9.52
a
dx
x
a
x
0
2
2
2
9.53
1
0
1
ln
dx
x
9.54
3
2
2
a
a
x
a
dx
9.55
1
0
2
1
x
dx
9.56
3
1
2
ln
1
e
x
x
dx
9.58
4
/
2
/
3
3
sin
cos
x
xdx
9.58
2
/
2
0
3
2
1
x
dx
9.59
dx
x
x
e
1
ln
sin
9.60
3
ln
2
ln
2
x
ch
dx
9.61
1
0
8
3
1
dz
z
z
9.62
3
/
6
/
4
2
sin
cos
x
x
dx
9.63
3
2
2
4
5
x
x
dx
9.64
2
1
1
x
x
e
dx
e
9.65
3
/
6
/
4
xdx
tg
9.66
2
0
1
2
1
2
dx
x
x
9.68
3
2
ln
e
e
x
x
dx
9.68
3
/
4
/
2
2
1
1
dx
tgx
x
tg
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