|
8.18
dx
x
x
2
2
1
8.28
dx
x
arcrgx
2
1
8.19
dx
x
x
5
2
4
8.29
dx
x
x
3
ln
8.20
x
x
dx
ln
8.30
xdx
x
2
2
1
“C” guruh
I. Integrallarni toping.
8.31
а
)
;
9
25
2
x
dx
в
)
;
1
4
x
x
dx
с
)
d
)
3
2
3
2
cos
x
x
dx
;
e
)
2
2
4
x
dx
x
8.32
a
)
b
)
dt
t
t
3
2
3
1
8.33
8.34.
2
16
d
8.35
dy
y
y
y
y
2
2
1
1
8.36.
dx
x
x
4
16
8.37.
8.38.
5
2
x
dx
8.39.
9
16
2
x
dx
840.
dx
x
x
x
6
3
2
1
1
8.41.
dx
e
x
x
1
5
8.42.
dx
x
x
2
3
2
3
8.43.
1
2
x
x
dx
8.44.
dx
x
x
x
4
2
2
4
2
2
8.45.
dx
x
x
4
6
4
2
2
8.46.
1
2
4
x
dx
x
8.47.
dx
x
x
1
1
3
8.48.
dx
x
2
sin
2
8.49.
xdx
ctg
2
8.50.
xdx
cth
2
8.51.
dx
x
x
cos
cos
5
8.52.
dx
e
e
x
x
2
4
2
8.53.
x
x
dx
2
cos
sin
2
8.54.
dx
x
x
x
2
2
2
1
5
1
8.55.
dx
x
x
x
4
2
8
6
3
8.56.
dx
e
e
x
x
5
8.57.
dx
x
x
x
1
1
8.58.
dx
tgx
ctgx
1
3
1
3
8.59.
dx
x
x
2
cos
1
cos
2
1
2
8.60.
dx
x
x
x
Eng sodda kasrlarni integrallash. Ratsional kasrlarni sodda
kasrlarga ajratish. Ratsional funksiyalarni integrallash
algoritmi
.O’zgaruvchini almashtirish usuli bilan integrallash.
Bo’laklab integrallash
.
Ratsional kasrlarni integrallash
O’zgaruvchini almashtirish (o’rniga qo’yish) usuli bilan integrallashning mohiyati shundan iboratki,
dx
x
f
integralni asosiy integrallash formulalarining birortasi oson integrallanadigan
du
u
F
integralga keltirishdan
iboratdir. Faraz qilamiz,
u
x
o’rganiladigan oraliqda uzluksiz, differensiallanuvchi funksiya bo’lsin, u holda
3
3
2
2
2
2
x
x
dx
x
dx
x
x
x
1
5
2
3
dx
x
x
1
5
3
2
3
2
4
9
x
dx
du
u
F
du
u
u
f
dx
x
f
. (8.25)
Yangi o’zgaruvchi
u
ga nisbatan integral topilgandan so’ng
x
u
o’rniga qo’yish yordamida uni
x
o’zgaruvchiga keltiriladi.
Bo’laklab integrallash
.
du
ud
u
d
tenglikning ikkala tomonini integrallab, quyidagini hosil qilamiz:
du
ud
u
du
ud
u
d
;
bu yerdan
du
u
ud
(8.26)
Ratsional kasrlarni integrallash
. Rastsional kasr deb
x
Q
x
P
m
n
/
ko’rinishidagi kasrga aytiladi, bu yerda
x
P
n
va
x
Q
m
mos ravishda
n
va
m
darajali ko’phadlar. Agar
m
n
bo’lsa, ratsional kasr to’g’ri ,
m
n
bo’lsa noto’g’ri kasr deyiladi. Har qanday noto’g’ri ratsional kasrni maxrajga suratni bo’lish orqali ko’phad to’g’ri
rasional kasr yig’indisi ko’rinishida tasvirlash mumkin. Shuning uchun ratsional kasrlarni integrallash to’g’ri ratsional
kasrlarni integrallashga keltiriladi. To’g’ri ratsional kasrni integrallash uchun uni eng sodda ratsionallar yig’indisi
ko’rinishida
k
ek
e
k
k
x
A
x
A
x
A
x
A
x
A
x
Q
x
P
1
1
1
1
1
2
1
12
1
11
...
...
...
bu yerda
va
ek
A
A
A
,...
,
12
11
-o’zgarmas haqiqiy sonlar;
k
- butun musbat sonlar
Eng sodda ratsional kasrlardan tashkil topgan integrallarni hisoblash.
1)
Birinchi turdagi eng sodda ratsional kasrlar:
C
a
x
A
dx
a
x
A
ln
2)
Ikkinchi turdagi eng sodda ratsional kasrlar:
2
;
1
1
1
n
N
n
C
a
x
n
A
dx
a
x
A
n
n
3)
Uchinchi turdagi eng sodda ratsional kasrlar:
0
2
;
2
;
0
4
;
2
2
2
2
2
2
2
2
2
p
q
a
p
x
t
q
p
dt
a
t
Ap
B
dt
a
t
At
dx
q
px
x
B
Ax
4)
To’rtinchi turdagi eng sodda ratsional kasrlar:
0
2
;
2
;
0
4
;
2
;
;
2
2
2
2
2
2
2
2
2
p
q
a
p
x
t
q
p
n
N
n
a
t
dt
Ap
B
a
t
tdt
A
dx
q
px
x
B
Ax
n
n
n
5)
Beshinchi turdagi eng sodda ratsional kasrlar:
2
;
1
2
1
1
2
2
2
2
n
C
a
t
n
a
t
tdt
n
n
6)
Oltinchi turdagi eng sodda ratsional kasrlar:
2
;
1
2
1
1
1
1
2
1
1
2
2
2
1
2
2
2
2
2
n
a
t
dt
n
a
a
t
n
a
a
t
dt
n
n
n
“A” guruh
O’zgaruvchilarni almashtirish usuli yordamida integrallarni toping. 8.61
а
)
в
)
dx
x
x
arctg
2
3
1
с
)
3
2
1
x
dx
x
d
)
dx
x
x
arctg
2
3
1
e
)
dx
x
x
x
1
ln
8.62.
а
)
dx
x
x
20
cos
20
sin
в
)
dx
x
7
7
11
dx
x
x
x
6
6
5
5
4
8.63.
x
arctg
arctgx
d
2
8.64.
2
/
3
2
1
sin
ctgx
x
dx
8.65.
dx
x
5
3
8.66.
2
3
2
x
dx
8.67.
dx
x
7
3
4
8.68.
dx
x
1
2
8.69.
dx
x
x
ctg
2
2
cos
1
3
8.70.
x
x
dx
2
ln
9
8.71.
5
2
x
xdx
8.72.
dx
x
x
4
1
5
2
8.73.
x
x
dx
arcsin
1
2
8.74.
dx
e
e
x
x
cos
8.75.
x
d
x
2
2
2
ln
1
ln
1
sec
8.76.
3
/
1
2
x
dx
x
8.77.
dx
x
x
x
x
2
4
3
2
8.78.
dz
z
z
ln
8.79.
x
x
e
dx
e
6
3
1
8.80.
dx
x
ctgx
x
2
sin
cos
5
8.81.
dx
e
x
x
/
1
2
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