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Trigonometrik funksiyalardan tashkil topgan ifodalarni integrallash
. Bu integrallarni
dx
x
x
R
cos
,
sin
(bu
yerda R – ratsional funksiya). Integrallash
x
x
tg
u
2
universal trigonometrik almashtirish
yordamida ratsional kasrlarni integralalshga keltiriladi.
U holda
2
2
2
2
1
2
,
1
1
cos
,
1
2
sin
u
du
dx
u
u
x
u
u
x
va
2
2
2
2
1
2
1
1
;
1
24
cos
;
sin
u
du
u
u
u
R
dx
x
x
R
bo’ladi.
“A” guruh
Trigonometrik funksiyalar qatnashgan ba’zi ifodalarni integrallarini hisoblang.
8.179.
а
)
x
dx
cos
5
3
b)
x
dx
3
cos
8.180.
а
)
x
x
xdx
cos
sin
sin
b
)
x
xdx
cos
1
cos
8.181.
ctgx
dx
1
8.182.
dx
x
x
x
x
cos
sin
cos
sin
8.183.
2
cos
sin
x
x
dx
8.184.
x
x
dx
sin
cos
1
8.185.
x
x
x
dx
cos
sin
5
sin
2
8.186.
x
x
dx
sin
1
cos
8.187.
x
x
xdx
2
2
sin
cos
6
sin
8.188.
x
dx
x
x
2
cos
sin
sin
3
8.189.
x
tgx
x
dx
2
2
cos
3
2
sin
8.190.
dx
tgx
tgx
1
1
8.191.
dx
x
x
2
3
cos
1
sin
8.192.
x
dx
cos
5
3
8.193.
x
x
dx
cos
7
sin
4
8
8.194.
tgx
dx
3
4
8.195.
x
x
xdx
3
3
cos
sin
cos
8.196.
dx
x
tgx
sin
1
Sinus va kosinuslarni ko’paytmasi vа darajasi qatnashgan ba’zi ifodalarni integrallarini toping.
8
.197.
xdx
2
cos
3
8.198.
xdx
5
cos
2
8.199.
xdx
x
2
2
cos
sin
8.200.
xdx
x
3
2
cos
2
sin
8.201.
xdx
7
cos
8.202.
xdx
6
sin
8.203.
dx
x
x
4
3
cos
sin
8.204.
x
x
dx
2
2
cos
sin
8.205.
dx
x
x
2
cos
2
sin
4
2
8.206.
xdx
x
3
2
cos
sin
8.207.
xdx
x
5
sin
3
cos
8.208.
dx
x
x
4
sin
2
sin
8.209.
dx
x
x
4
cos
2
5
sin
8.210.
xdx
tg
2
8.211.
xdx
x
x
3
sin
2
sin
sin
8.212.
dx
x
x
3
2
sin
cos
8.213.
xdx
ctg
3
8.214.
xdx
tg
3
6
8.215.
d
tg
4
4
sec
8.216.
x
xdx
ctg
4
2
/
3
sin
8.217.
x
xdx
4
sin
2
cos
8.218.
x
x
dx
cos
sin
5
“B” guruh
Ba’zi bir irratsional ifodalar qatnashgan integrallarni toping.
8.219.
а
)
dx
x
x
1
2
b
)
dx
x
x
3
/
1
1
2
8.220.
8.221.
dx
x
x
x
3
2
1
8.222.
x
dx
x
x
3
8.223.
2
3
1
1
1
x
dx
x
x
8.224.
dx
x
x
25
8.225.
dx
x
x
x
x
x
1
1
3
8.226.
dx
x
x
x
x
6
7
4
5
3
8.227.
dx
x
x
1
1
8.228.
3
6
1
x
dx
x
8.229.
dx
x
x
x
x
x
3
6
3
2
1
8.230.
6
3
2
2
2
x
x
x
dx
8.231.
dx
x
x
3
4
1
8.232.
3
2
3
1
x
dx
x
8.233.
dx
x
x
3
6
2
1
dx
x
x
2
/
3
1
8.234.
dx
x
x
2
5
4
8.235.
3
3
1
x
dx
8.236.
dx
x
x
2
3
2
1
8.237.
2
7
1
x
dx
x
8.238.
1
2
4
x
x
dx
8.239.
4
/
1
3
1
x
x
dx
“C” guruh
Eyler аlmashtirish yordamida integrallarni toping.
8.240.
1
4
2
x
x
x
dx
8.241.
4
2
x
x
x
dx
8.242.
2
2
8
6
x
x
xdx
8.243.
5
2
2
2
x
x
dx
x
8.244.
5
2
x
x
x
dx
8.245.
1
2
x
x
dx
8.246.
3
2
10
7
2
x
x
x
dx
JAVOBLAR
8.31.
a
) Integrallash jadvali (8.18) formuladan foydalansak,
C
x
x
dx
x
dx
y
5
3
arcsin
3
1
9
/
25
3
1
9
25
2
2
b
) Integralni surat va maxrajini
1
4
x
x
ga ko’paytirib,
uni jadval integraliga olib kelamiz.
C
x
x
C
x
x
dx
x
x
dx
x
x
x
x
x
x
x
x
dx
x
x
x
x
dx
y
2
/
3
2
/
3
2
/
3
2
/
3
1
15
2
4
15
2
1
3
2
4
3
2
5
1
1
4
5
1
1
4
1
4
1
4
1
4
1
4
1
4
c
) Integral osti ifodaning suratini
3
3
2
2
2
2
x
x
x
ko’rinishida yozamiz, u holda
C
x
x
x
x
dx
x
dx
dx
x
x
x
x
dx
x
x
x
y
1
3
3
ln
3
2
1
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
;
d
)
x
x
2
cos
2
2
cos
1
va
x
x
2
2
cos
1
sin
ko’rinishidagi trigonometrik ayniyatlardan foydalansak, u holda
C
x
tgx
dx
x
dx
dx
x
x
x
dx
dx
x
x
dx
x
x
2
3
cos
2
3
cos
cos
1
cos
2
1
cos
2
sin
2
1
2
cos
1
sin
2
1
2
2
2
2
2
2
2
;
e
)
Kasr suratiga 4 ni qo’shib ayiramiz, u holda
C
x
arctg
x
x
x
dx
dx
x
x
x
x
dx
x
2
2
4
2
3
2
4
2
2
4
4
2
3
2
2
2
4
2
4
8.32
a
)
C
x
x
x
1
7
2
2
3
b
)
C
t
t
t
3
2
3
5
2
8.33
C
x
x
2
/
5
6
/
19
5
2
19
30
8.34
C
4
arcsin
8.35 .
C
y
y
y
y
2
1
2
5
2
2
8.36
C
x
x
x
4
3
2
8.37.
C
x
3
2
arcsin
2
1
8.38.
C
x
x
5
ln
2
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