ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
1185
BA'ZI XUSUSIY INTEGRAL TENGLAMALAR VA ULARNI YECHISH
Djurayev Yuldosh Xurramovich
Denov tadbirkorlik va pedagogika instituti 2-kurs magistri
https://doi.org/10.5281/zenodo.7978017
Annotatsiya.
Mazkur maqolada ba'zi xususiy integral tenglamalar va ularni yechish,
shuningdek aniq integralning ta`rifi hamda uning geometrik ma`nolari haqida so’z boradi.
Maqola mavzusi misollar yordamida ochib berilgan.
Kalit so’zlar:
xususiy integral tenglamalar, tenglamalar va ularni yechish, integralning
ta`rifi.
SOME PARTICULAR INTEGRAL EQUATIONS AND THEIR SOLUTION
Abstract.
This article talks about some particular integral equations and their solution, as
well as the definition of definite integral and its geometric meanings. The topic of the article is
explained with the help of examples.
Key words:
special integral equations, equations and their solution, definition of integral.
НЕКОТОРЫЕ ЧАСТНЫЕ ИНТЕГРАЛЬНЫЕ УРАВНЕНИЯ И ИХ
РЕШЕНИЕ
Аннотация.
В этой статье говорится о некоторых частных интегральных
уравнениях и их решении, а также об определении определенного интеграла и его
геометрических смыслах. Тема статьи поясняется с помощью примеров.
Ключевые слова:
специальные интегральные уравнения, уравнения и их решение,
определение интеграла.
Aniq integral- matematik analizning asosiy tushunchalaridan biridir. Egri chiziqlar bilan
chegaralangan yuzalarni, egri chiziq yoylari uzunliklarini, hajmlarini, ishlarni, tezliklarni,
yo’llarni, inersiya momentlarini hisoblash masalasi u bilan bog’liq.
[a,b]
kesmada
y=f(x)
uzluksiz funksiya berilgan bo’lsin. Quyidagi amallarni bajaramiz.
1)
[a,b]
kesmani
a= x
0
,x
1
,x
2
,....,x
n-1
,x
n
=b
nuqtalar bilan
n
ta qismga ajratamiz va ular
quyidagicha joylashgan bo’lsin.
a= x
0
<x
1
<x
2
<....<x
n-1
<x
n
=b
Bularni qismiy intervallar deymiz.
1
2
3
n
a=x
0
x
1
x
2
x
3
x
n-1
x
n
=b õ
2)
Qismiy intervallarning uzunliklarini quyidagicha belgilaymiz:
x
1
=x
1
-x
0
;
x
2
=x
2
-x
1
;
x
3
=x
3
-x
2
;.......
x
i
=x
i
-x
i-1
;....
x
n
=x
n
-x
n-1
;
3)
Har bir qismiy intervalning ichidan bittadan ixtiyoriy nuqta olamiz:
1
,
2
,
3
,......
n-1
,
n
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
1186
4)
Olingan
nuqtalarda funksiyaning qiymatini topamiz:
f(
1
); f(
2
);f(
3
),...... f(
n-1
); f(
n
)
5)
Har bir funksiyaning hisoblangan qiymatini tegishli qismiy intervalning uzunligiga
ko’paytiramiz:
f(
1
)
x
1
; f(
2
)
x
2
; f(
3
)
x
3
,...... f(
n
)
x
n
6)
Hosil bo’lgan ko’paytmalarni qo’shamiz va
deb belgilaymiz.
=
f(
1
)
x
1
+ f(
2
)
x
2
+f(
3
)
x
3
+..... + f(
n-1
)
x
n-1
+f(
n
)
x
n
;
Shunday qilib, hosil bo’lgan
yig’indi
f(x)
funksiya uchun
[a,b]
kesmada tuzilgan integral
yig’indi deb ataladi va quyidagicha belgilanadi.
i
i
n
i
x
f
)
(
1
(1)
Bu integral yig’indining geometrik ma`nosi, agar
f x
( )
0
bo’lsa, u holda asoslari
x
1
,
x
2
,...
x
n
va balandliklari
f(
1
), f(
2
),... f(
n
)
bo’lgan to’g’ri to’rtburchak yuzlarining yig’indisidan
iborat.
Agarda bo’lishlar sonini,
n
ni orttira borsak (
n
)da u holda eng katta intervalning
uzunligi nolga intiladi, ya`ni max
0
i
x
bo’ladi.
Ta`rif:
Agar
S
integral yig’indi
[a,b]
kesmani qismiy
[x
i-1
, x
i
]
kesmalarga ajratish usuliga
va ularning har biridan
1
nuqtasini tanlash usuliga bog’liq bo’lmaydigan chekli songa intilsa, u
holda shu son
[a,b]
kesmada
f(x)
funksiyadan olingan aniq integral deyiladi va quyidagicha
belgilanadi.
f x dx
a
b
( )
f(x)
dan
x
bo’yicha
a
dan
b
gacha olingan aniq integral deb o’qiladi.
Bu yerda
f(x)
integral ostidagi funksiya
[a,b]
kesma-integrallash oralig’i;
a
son integralning quyi
chegarasi,
b
son integralning yuqori chegarasi;
Shunday qilib, aniq integralning ta`rifidan quyidagini yozish mumkin.
i
i
n
i
x
b
a
x
f
dx
x
f
i
)
(
)
(
1
0
max
lim
Aniq integral hamma vaqt mavjud bo’lavermas ekan. Aniq integralning mavjudlik
teoremasini quyida keltiramiz. (Isbotsiz).
Teorema:
Agar
f(x)
funksiya
[a,b]
kesmada uzluksiz bo’lsa, u integrallanuvchidir, ya`ni
bunday funksiyaning aniq integrali mavjuddir.
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
1187
Shunday qilib,
f x dx
a
b
( )
aniq integralning qiymati
y=f(x)
funksiyaning grafigi bilan va
x=a, x=b
to’g’ri chiziqlar bilan chegaralangan egri chiziqli trapetsiyaning yuziga son jihatdan teng
bo’ladi.
1-
Izoh: Aniq integralning chegaralari almashtirilsa, integralning ishorasi o’zgaradi.
a
b
b
a
dx
x
f
dx
x
f
)
(
)
(
2-Izoh. Agar aniq integralning chegaralari teng bo’lsa, har qanday funksiya uchun quyidagi
tenglik o’rinli;
f x dx
a
а
( )
0
haqiqatdan ham, geometrik nuqtai nazardan egri chiziqli trapetsiya asosining uzunligi nolga teng
bo’lsa, uning yuzi ham nolga teng bo’ladi.
Aniq integralning asosiy xossalari
1- xossa: O’zgarmas ko’paytuvchini aniq integral belgisining tashqarisiga chiqarish
mumkin.
Аf x dx
А f x dx
a
b
a
b
( )
( )
Isbot:
b
a
i
i
n
x
i
i
n
x
b
a
dx
x
f
A
x
f
A
x
Af
dx
x
Аf
i
i
)
(
)
(
)
(
)
(
1
0
0
max
1
0
0
max
lim
lim
2-xossa: Bir necha funksiyalar algebraik yig’indisining aniq integrali qo’shiluvchilar aniq
integrallarning algebraik yig’indisiga teng.
Masalan:
b
a
b
a
b
a
dx
x
f
dx
x
f
dx
x
f
x
f
)
(
)
(
)
(
)
(
2
1
2
1
3-xossa. Agar
[a, b]
kesmada
f(x)
va
(x)
funksiyalar uchun
f(x)
(x)
shart bajarilsa, u
holda
f x dx
x dx
a
b
a
b
( )
( )
bo’ladi.
4-xossa: Agar
[a,b]
kesma bir necha qismga bo’linsa, u holda
[a,b]
kesma bo’yicha aniq
integral har bir qism bo’yicha olingan aniq integrallar yig’indisiga teng.
Masalan:
a<c< b
bo’lsa, u holda
f x dx
f x dx
f x dx
a
b
a
с
с
b
( )
( )
( )
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
1188
5-xossa: Aniq integralning qiymati funksiyaning ko’rinishiga va integrallash chegaralariga
bog’liq, lekin integral ostidagi ifodaning harflariga bog’liq emas.
f x dx
f t dt
f z dz
a
b
a
с
a
b
( )
( )
( )
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M. Rahman. Integral Equations and Their Applications, WIT press, Southampton, Boston,
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