Eyler integrallarining tatbiqlari

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SCIENTIFIC PROGRESS


VOLUME 2 I ISSUE 1 I 2021
ISSN: 2181-1601


1
r(4)r(5) 3!* 4! 1
I x³(1 — x)⁴dx = B(4,5) = ———— = ——— = ——
r(9) 8! 280
0


Shunday qilib, integralning qiymati — ga teng ekanligini topdik.
280


3- Misol.


n
2

1. 
   ~ⁿ2^{x COS}2^xdx
   

0


integralni hisoblaymiz.


Yechilishi. Bu integralning akademik litsey kursida hisoblash uchun litsey


o'quvchisi ikkilangan burchak sinusi formulasidan foydalanib, integral ostida berilgan


ifodani bitta funksiyaga keltiradi va darajani pasaytirish formulasini qo'llab, hisoblaydi.


n
2
M2* _C0S2^xdx =
0


n n

2. 
   2
   

¹ / sin²2xdx =¹ / 1 - ^C°^S4^X dx = ¹ [(x — 1sin4x)]2 = £
4 4 2 8 4 ⁰ 16
00


2- usul. Endi berilgan integralni Eyler integrallari yordamida hisoblaymiz. Bunda


quyidagi almashtirishlarni bajaramiz:


n
2


/^sin2^{x C}n^s2xdx =
0


H
^Si^nx = ^y,^dy = ^COSxdx,^x = 0,^y = ^0;x = 2,^y = 1


1
/ y²^1 — y² dy
0


integralni hosil qilamiz Bu integralda yana almashtirish usulini qo'llab, berilgan


integralning qiymatini topamiz, ya'ni


1
/ y²^1 — y²dy = y² = t, dy =
0


1
— dt ,y = 0,t = 0;y = 1,t = 1 2Vt


-1f_ti_{(1 t)}1_dt -M^{3 3}) ^{ir2(3) 1 {}1^r(2)} = *
2 J ^ ⁽ ) 2^B (2,2) 2 r(3) 2 2! 16
0
n
Demak, integrakning qiymati —


4-Misol. y = x² va y = x³ chiziqlar bilan chegaralangan shaklning yuzini


topamiz.
Yechilishi. Berilgan egri chiziqlarni tenglashtirib, kesishish nuqtalarini topamiz va grafigini yasaymiz.
x² = x³, x²(x — 1) = 0 ,x = 0,x = 1.


Ma'lumki, ta'lab qilingan shaklning yuzi aniq integral bilan topiladi.


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SCIENTIFIC PROGRESS


VOLUME 2 I ISSUE 1 I 2021
ISSN: 2181-1601


1
1 1 11 1
S = J(x^2- x³⁾dx = ^[(-x3 ^- ~4)⁴)]o = 3 — 4 = ^
0
Endi bu integralni Eyler integrallari bilan hisoblaymiz.
11
r(3)r(2) 2! 1 1
/ (x² - x³⁾dx = / x²⁽1 - x)dx = 5(3,2) = = — = —
00
1
Demak, shaklnmg yuzi — kv. bir
5-Misol.


1
R^(x-1)2^dx
0
Integralni hisoblang.
Yechilishi. 1-usul. Bu usulda integral akademik litsey programmasida


quyidagicha topiladi.
1
^/vx(x-1)2^dx
0


1
= /V7(^x2
0


2 7 4 5 2 3
- 2x + 1)dx = [(-x2 - -x2 + -x2)]0
/ uJ O


2 4 2 _ 16
7-5+ 3 = 105


2-usul. Berilgan integral Eyler integrallari orqali quyidagicha hisoblanadi ya’ni


1
P^(x-I)2dx =


1
/ x3^-1(1 - x)^3-1dx =)5(|, 3)


Endi r (n + 1) = 1*³*⁵*-*⁽²ⁿ 1)^~, B(a, b) = ^r(^)^r(b) formulalami qo'llab, 2 2ⁿ , ^v , ⁷ rfo+b) ^,
B(3, 3) funksiyaning qiymatini topamiz.
3 ^r(l)^r(³) ^{r(1 + 1})^r(³) 2^*2! 16
^², ' r(3+3) r(4 + 1) ^* ³₂*₄⁵ * ⁷V^ ¹⁰⁵
16
Demak, berilgan misolning javobi; —


MUHOKAMA
Ushbu Eyler integrallarining afzalliklari: matematik bilimlarini chuqurlashtirish, fikrlash doirasini kengaytirish, tasavvur qobiliyatini o’stirish, Eyler integrallari hisoblangan beta va gamma funksiyalarning muhim xossalarini o’rganib ularni turli xil integrallarni hisoblashga tadbiq qilishga, fizika va mexanikaning ba’zi masalalarini


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SCIENTIFIC PROGRESS VOLUME 2 I ISSUE 1 I 2021
ISSN: 2181-1601


yechishga hamda adabiyotlarda berilgan bir qator murakkab integrallarni hisoblashga o’rgatadi.
Kamchiliklari: Eyler integrallarining tatbiqlari qisqa yoritildi.
NATIJA
Ushbu Eyler integrallarining tatbiqining afzalliklari: talabalarning Eyler integrallarining muhim xossalarini tahlil qilish va isbotlashga, Eyler integrallarining akademik litsey kursidagi integrallarni hisoblash tadbiq qilishga imkon yaratadi, fikrlash doirasini kengaytiradi, tasavvurini o’stiradi hamda fanga nisbatan qiziqishini oshiradi.Natijada fan yuzasidan bilimlari yanada mustahkamlanadi.
Metodning kamchiliklari deyarli aniqlanmagan. Faqat o’qituvchi va o’quvchidan ozgina izlanish talab qilinadi.
XULOSA
Ma’lumki, hozirgi vaqtda mamlakatimiz Prezidenti tomonidan matematika fani va uni amaliyotga qo’llashni rivojlantirishga katta ahamiyat berilib, bir qator qarorlar imzolangan.Qarorlar ijrosini ta’minlashning negizida albatta fanni talabalarga qulay matematik usullardan foydalanib o’rgatish yotadi. Maqolada Eyler integrallarining akademik litsey kursidagi integrallarni hisoblashga qo’llash bo’yicha izlanishlar olib borilgan.
Eyler integrallari fizika va mexanikaning ba’zi masalalariga tadbiq qilinsa talabalarga qulaylik tug’diradi.
REFERENCES

1. 
   PacynoB X.P., PamugoB A.ffl. (2020). OpraHU3a^ua npakTUHeckoro 3aHamua Ha ocHOBe nnnoBannoniibix TexHonorun Ha ypokax MameMamuku. HayKa, mexHUKa u o6pa3OBaHue, 8(72), 29-32.
   
2. 
   PacynoB X.P., PaynoBa M.X. (2021). Pont MameMamuku b 6uonoruneckux Haykax. npodneMbi nedaroruKU, 2(53), 7-10.
   
3. 
   PacynoB T.\, PacynoB X.P. (2021). Y3rapumu nerapanaHraH ^yHK^uanap 6ynuMUHu y^uTumra goup Memoguk TaBCuanap. Scientific progress, 1(2), 559-567.
   
4. 
   yMapoBa y.y. (2020). Pohb coBpeMeHHbix uHTepakTUBHbix MeTogoB b u3yneHuu TeMbi «Mno>ihumu». BecmHUK HayKU u o6pa3OBaHux, 16(94), 21-24.
   
5. 
   Boboeva M.N., Rasulov T.H. (2020). The method of using problematic equation in teaching theory of matrix to students. Academy, 4(55), 68-71.
   
6. 
   Rasulov T.H., Rashidov A.Sh. (2020). The usage of foreign experience in effective organization of teaching activities in Mathematics. International Journal of Scientific & Technology Research, 4(9), 3068-3071.
   

Uzbekistan


www.scientificprogress.uz


Page 1404




SCIENTIFIC PROGRESS VOLUME 2 I ISSUE 1 I 2021
ISSN: 2181-1601

7. 
   Mardanova F.Ya., Rasulov T.H. (2020). Advantages and disadvantages of the method of working in small group in teaching higher mathematics. Academy, 4(55), 65-68.
   
8. 
   TomeBa H.A. (2020). Me^guc^unnuHapHbie cb^3u b npenogaBaHuu KOMnnekCHoro aHanu^a. BecmHUK HayKU u o6pa3OBaHux, 16(94), 29-32.
   
9. 
   XanuTOBa X.r. (2020). Hcnonb3OBaHue ^BpucTuneckoro Memoga npu o6tacHeHuu TeMbi «1 le'iipe'pbiBiibie' .iHiie'HHbie' one'paropbi» no npegMemy «OyHK^uoHanbHbiH
   

aHanu3». BecmHUK HayKU u o6pa3OBaHux, 16(94), 25-28.

10. 
    Rasulov T.H., Rasulova Z.D. (2019). Organizing educational activities based on interactive methods on mathematics subject. Journal of Global Research in Mathematical Archives, 6(10), 43-45.
    
11. 
    PacynoB X.P., H>icohhu enuMnapu xaKuga, Scientific progress, 1(2), 455-462.
    
12. 
    PacynoB X.P., PamugoB A.ffl. (2020). O cyinecTBOBanmi oooomennoro pemeHua kpaeBOH 3aganu gna HenuHeHHoro ypaBHeHua CMemaHHoro muna, BecmHUK HayKU u o6pa3OBaHun, 19-1 (97), 6-9.
    
13. 
    Dilmurodov E.B. (2020). Discrete eigenvalues of a 2x2 operator matrix. ArXiv:2011.09650. 1-12.
    
14. 
    Rasulov T.H., Dilmurodov E.B. (2020). Analysis of the spectrum of a 2x2 operator matrix. Discrete spectrum asymptotics. Nanosystems: Phys., Chem., Math., 2(11), 138- 144.
    
15. 
    PacynoB T.X., ^unMypogoB ^.E. (2020). EeckOHenHOCTb nucna coocTBe'iiiibix 3HaneHUH one'paropnbix (2x2)-MaTpu^. AcuMnTOTUka guckpeTHoro cnekTpa. TM&. 3(205), 368-390.
    
16. 
    Dilmurodov E.B. (2019). On the virtual levels of one family matrix operators of order 2. Scientific reports of Bukhara State University, 1, 42-46.
    
17. 
    ^unMypogoB ^.E. (2017).Huchoboh o6paa MHoroMepHOH ooooine'iinoH Mogenu Opugpuxca. Monodou yueuuiu, 15, 105-106.
    
18. 
    ^unMypogoB ^.E. (2016).KBagparHHiibiH huchoboh o6paa ogHOH 2x2
    

onepaTopHOH Marpniibi. Monodou yueuuiu, 8, 7-9.

19. 
    Rasulov T.H., Dilmurodov E.B. (2019). Threshold effects for a family of 2x2 operator matrices. Journal of Global Research in Mathematical Archives, 10(6), 4-8.
    
20. 
    PacynoB T.X., ^unMypogoB ^.E. (2014). HccnegoBaHue huchoboh o6nacru
    

3HaneHUH ogHOH onepaTopHOH Marpnnbi. BecmH. CaMapcK. rocyd. mexH. yH-ma. Cep. (bir;.-Mcim. HayKU., 35 (2), 50-63.

21. 
    Rasulov T.Kh., Dilmurodov E.B. (2015). Estimates for quadratic numerical range of a operator matrix. Uzbek Math. Zh., 1, 64-74.
    
22. 
    ^unMypogoB ^.E. (2018). CnekTp u KBagparuHHbm huchoboh o6pa3 o6o6me'nnoH Mogenu Opugpuxca. Mocwdou yueuuiu, 11, 1-3.
    

Uzbekistan


www.scientificprogress.uz


Page 1405

SCIENTIFIC PROGRESS VOLUME 2 I ISSUE 1 I 2021

ISSN: 2181-1601

23. 
    PacynoB T.X., EaxpoHOB E.H. (2015). O cnekmpe TeH3opHoH cyMMb Mogenen Opugpuxca, Monodou yueHuu, 9, 17-20.
    
24. 
    .HaKaeB C.H., PacynoB T.X. (2003). Mogent b meopuu BO3\iyineiiHH cymecTBeinioro cnekmpa MHoronacTUHHbix onepamopoB. MameM. 3aMemKU., 4(73), 556-564.
    
25. 
    HakaeB C.H., PacynoB T.X. (2003). 06 ^$$ekTe E^uMOBa b Mogenu Teopuu BO3\iyineiiHH cyine'CTBe'inioro cnekTpa. (lcyiiKiiuoiicuibiimii ciiicuii; u ero npuooom, 1(37):1, 81-84.
    
26. 
    Albeverio S., Lakaev S.N., Rasulov T.H. (2007). On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics. Journal of Statistical Physics, 2(127):2, 191-220.
    
27. 
    Albeverio S., Lakaev S.N., Rasulov T.H. (2007). The Efimov Effect for a Model Operator Associated with the Hamiltonian of non Conserved Number of Particles. Methods of Functional Analysis and Topology, 1(13), 1-16.
    
28. 
    ABe3OB A.X. (2018). HccnegoBaHue bhu^hu^ cooTHomeHua cropoH np^MoyronbHoro conna Ha napa\ie'Tpb gu$$y3uoHHoro ^akena. yueubiu XXI BeKa, 1(4), 4-5.
    
29. 
    ABe3OB A.X. (2017). Bb6op MaTeMaTuneckOH Mogenu u uccnegoBaHue
    

Tpewiepiiibx Typ6yneHTHbix crpyH. Monodouyueuuu, 15, 101-102.

30. 
    ABe3OB A.X. (2016). I le'KOTopue' 'iHcjeiinbe peayjbTaTb uccnegoBaHua
    

Tpe'wie'piiux Typoyje'HTiibx crpyfi peaiTipyioiHHx ra3OB. Monodouyueuuu, 12, 1-2.


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