|
Click here to buy
A
B
B
Y
Y
PD
F Transfo
rm
er
2
.0
w
w
w .A
B B Y Y.
c o
m
Click here to buy
A
B
B
Y
Y
PD
F Transfo
rm
er
2
.0
w
w
w .A
B B Y Y.
c o
m
66
I N T E G R A L L A R
1. N'yuton-Leybnis formulasi:
( )
( )
( )
( ).
b
b
a
a
f x dx
F x
F b
F a
S
=
=
=
-
ò
2. Egri chiziq bilan chegaralangan yuzalarni hisoblash:
a) Egrichiziqli trapesiya yuzi:
( )
b
a
S
f x dx
=
ò
;
b) agar
1
2
( )
( )
0
f x
f x
>
>
bo`lsa, u holda
[
]
1
2
( )
( )
;
b
a
S
f
x
f
x
d x
=
-
ò
bo`ladi.
3.
(
)
( ) ( )
0
y
f x
f x
=
>
egri chiziq
aylanganda hosil bo'lgan jism hajmi:
2
2
( )
.
b
b
a
a
V
f
x dx
y dx
p
p
=
=
ò
ò
4.
» : ( ),
AB y
f x
a
x
b
=
£ £
yoyning uzunligi:
2
1
( )
b
a
f
x dx
l
¢
=
+
ò
.
5.
»
( ),
:
( ),
x
x t
AB
y
y t
t
a
b
=
ì
í =
£ £
î
yoyning uzunligi:
2
2
( )
( )
x
t
y
t dt
l
b
a
¢
¢
=
+
ò
.
6.
(
)
( ) ( )
0
y
f x
f x
=
³
,
[ ]
,
x
a b
Î
egri chiziqni
OX
o`qi atrofida
aylantirishdan hosil bo'lgan aylanish sirtining yuzini topish:
2
0
2
( )
1
( )
b
f
x
f
x d x
S
p
¢
=
×
+
ò
.
Integrallash qoidasi
1.
(
)
(
)
,
.
b
b
a
a
k
f
x
d x
k
f
x
d x
k
c o n s t
×
=
=
ò
ò
2.
[
]
(
)
(
)
(
)
(
)
.
b
b
b
a
a
a
f
x
g x
d x
f
x d x
g x d x
+
=
+
ò
ò
ò
Click here to buy
A
B
B
Y
Y
PD
F Transfo
rm
er
2
.0
w
w
w .A
B B Y Y.
c o
m
Click here to buy
A
B
B
Y
Y
PD
F Transfo
rm
er
2
.0
w
w
w .A
B B Y Y.
c o
m
67
3.
( )
( )
( )
( )
( )
( ).
b
b
b
a
a
a
f x d g x
f x
g x
g x d f x
=
×
-
ò
ò
4.
(
)
1
(
) , 0,
`
.
b
b
a
a
f kx
c dx
k f kx
c
k
c
o zgarmas sonlar
¢
+
=
×
+
¹
-
ò
5. Agar
[
]
( )
( ),
;
,
0
f
x
f x
x
a a
a
- =
Î -
>
bo`lsa,
0
( )
2
( )
a
a
a
f x dx
f x dx
-
=
ò
ò
bo`ladi.
6. Agar
[
]
(
)
( ), ; , 0
f
x
f x
x
a a
a
- = -
Î -
> bo`lsa,
( )
0
a
a
f x dx
-
=
ò
bo`ladi.
7. Agar
[ ]
( ) 0,
;
f x
x
a b
³
Î
bo`lsa,
( )
0
b
a
f x d x
³
ò
bo`ladi.
8. Agar
( )
0
;
( )
0
a
x
c da
f x
c
x
b da
f x
< <
³
< <
<
bo`lsa,
( )
( )
( )
b
c
a
a
a
c
f x d x
f x d x
f x d x
=
-
ò
ò
ò
bo`ladi.
Aniqmas integral
1
1
1.
.
2.
.
1
m
m
dx
ln ln x
C
sin x cosxdx
sin
x
C
x lnx
n
+
=
+
×
=
+
×
+
ò
ò
(
)
(
)
2
2
2
1
1
3. 1
.
4.
1 .
1
1
1
x dx
ln x
x
C
x
ln x dx
x
C
x
a
a
a
a
a
+
æ
ö
= - - +
×
=
×
-
+
¹ -
ç
÷
ç
÷
+
+
-
è
ø
ò
ò
2
2
2
2
2
5.
.
2
2
x
a
x
a
x dx
a
x
arcsin
C
a
-
=
-
+
+
ò
(
)
2
1
6. 1
.
2
arctgx dx
x arctgx
ln
x
C
= ×
- ×
+
+
ò
(
)
7. 1
.
x
x
x e dx
x
e
C
×
=
- × +
ò
(
)
2
2
8.
2
2
.
x
x
x e dx
x
x
e
C
=
-
+
× +
ò
2
2
1
1
9.
2
.
10.
2
.
2
4
2
4
x
x
sin xdx
sin x
C
cos xdx
sin x
C
=
-
+
=
+
+
ò
ò
3
3
3
3
11.
.
12.
.
3
3
cos x
sin x
sin x dx
cos x
C
cos x dx
sinx
C
= -
+
+
=
-
+
ò
ò
1
13.
.
ln xdx
x ln x
ln
x dx
C
a
a
a
a
-
= ×
-
+
ò
ò
Click here to buy
A
B
B
Y
Y
PD
F Transfo
rm
er
2
.0
w
w
w .A
B B Y Y.
c o
m
Click here to buy
A
B
B
Y
Y
PD
F Transfo
rm
er
2
.0
w
w
w .A
B B Y Y.
c o
m
68
2
14. 1
.
arcsin x dx x arcsin x
x
C
= ×
+
-
+
ò
2
2
2
2
2
2
2
2
2
2
, 4 ,
4
4
15.
1
2
4
, 4 .
4
2
4
cx b
arctg
C
agar
b
ac
ac b
ac b
dx
a bx cx
cx b
b
ac
ln
C
agar
b
ac
b
ac
cx b
b
ac
+
ì
+
<
ï
-
-
ï
= í
+ +
+ -
-
ï
+
>
ï
-
+ +
-
î
ò
2
2
1
16.
2
2
,
0.
dx
ln cx
b
c a
bx
cx
C
c
c
a
bx
cx
=
+ +
+
+
+
>
+
+
ò
2
2
2
17.
4
cx b
a bx cx dx
a bx cx
c
+
+
+
=
+
+
-
ò
2
2
2
4
2
2
.
8
b
a c
ln
c x
b
c
a
b x
cx
C
c
-
-
+ +
+
+
+
2
2
2
2
1
2
4
1 8. .
4
2
4
dx
cx
b
b
ac
ln
C
a
bx
cx
b
a c
cx
b
b
a c
- +
+
=
+
+
-
+
-
+ +
+
ò
2
2
1
2
19.
,
0.
4
dx
cx b
arcsin
C
c
c
a
bx
cx
b
ac
-
=
+
>
+
-
+
ò
2
2
2
20.
4
cx
b
a
bx
cx dx
a
bx
cx
c
-
+
-
=
+
-
+
ò
2
2
2
4
2
.
8
4
b
ac
cx b
arcsin
C
c
b
ac
+
-
+
+
+
(
)(
) (
)
(
)
21. .
a
x
dx
a
x b
x
a b ln
a
x
b
x
C
b
x
+
=
+
+
+ -
+ +
+
+
+
ò
(
)(
) (
)
22. .
a
x
a
x
dx
a
x b
x
a b arcsin
C
b
x
a b
-
+
=
+
+
+
+
+
+
+
ò
(
)(
) (
)
23. .
a
x
b
x
dx
a
x b
x
a b arcsin
C
b
x
a b
+
-
= -
+
-
-
+
+
-
+
ò
2 4 .
,
.
s h x d x
ch x
C
ch x d x
s h x
C
=
+
=
+
ò
ò
25.
,
.
thxdx
lnchx
C
cthxdx
lnshx
C
=
+
=
+
ò
ò
(
)
(
)
(
)
(
)
26.
,
.
2
2
sin m
n x
sin m
n x
sin mx sin nx dx
C
m
n
m
n
m
n
+
-
×
= -
+
+
¹
+
-
ò
Click here to buy
A
B
B
Y
Y
PD
F Transfo
rm
er
2
.0
w
w
w .A
B B Y Y.
c o
m
Click here to buy
A
B
B
Y
Y
PD
F Transfo
rm
er
2
.0
w
w
w .A
B B Y Y.
c o
m
69
(
)
(
)
(
)
(
)
27.
,
.
2
2
sin m
n x
sin m
n x
cos mx cos nx dx
C
m
n
m
n
m
n
+
-
×
=
+
+
¹
+
-
ò
(
)
(
)
(
)
(
)
28.
,
.
2
2
cos m
n x
cos m n x
sin mx cos nx dx
C
m
n
m
n
m
n
+
-
×
= -
-
+
¹
+
-
ò
(
)
2
2
29. (
)
.
ax
ax
e
sin nx dx
e
a sin nx
n cos nx
a
n
C
×
=
×
- ×
+
+
ò
(
)
2
2
30. (
)
.
ax
ax
e
cos nx dx
e
a sin nx
n cos nx
a
n
C
×
=
×
+ ×
+
+
ò
2
2
2
2
2
, ,
2
31.
1
2
, .
2
a b
x
arctg
tg
C
agar a
b
a b
a
b
dx
x
b a tg
a b
a bcosx
ln
C
agar a
b
x
b
a
b a tg
a b
ì
æ
ö
-
×
+
>
ï
ç
÷
ç
÷
+
-
ï
è
ø
ï
= í
- ×
+
+
+
ï
+
<
ï
-
- ×
-
+
ï
î
ò
2
2
2
2
2
2
2
2
2
2
2
2
,
,
32.
s
1
2
, .
2
x
a tg
b
arctg
C
agar a
b
a
b
a
b
dx
x
a b inx
a tg
b
b
a
ln
C
agar a
b
x
b
a
a tg
b
b
a
ì
×
+
ï
+
>
ï
-
-
ïï
= í
+
×
+ -
-
ï
+
<
ï
-
ï
×
+ +
-
ïî
ò
Do'stlaringiz bilan baham: |
