Maple dasturi yordamida aniqmas va aniq integrallarni hisoblash



8
>convert(%,polynom);

1 1 _


1 _2 1 _


1 _4 17 _


1 _6 31 _ 1 _

₂  _{8 } x 
₂  _
 _{48 } x 
₂  _
^ 5760 ^_ ^{x }
₂  _
^ 80640 ^_ ^{x }
₂  _

>series(sum(i*x^5,i=1..12),x=1);

78  390 ( x  1 )  780 ( x  1 )²  780 ( x  1 )³  390 ( x  1 )⁴  78 ( x  1 )⁵

>convert(series(exp(x),x=ln(2),4),polynom);

2  2 x  2 ln( 2 )  ( x  ln( 2 ) )²  ¹ ( x  ln( 2 ) )³
3
Shunday funksiyalar borki ularning aniqmas integralini topishning iloji yo’q.
Masalan,

1
ln⁽𝑥^{) ,}
sin⁽𝑥⁾
,
𝑥
cos⁽𝑥⁾



𝑥

funksiyalar. Bunday funksiyalarning aniqmas integralini darajali qator ko’rinishida aniqlash mumkin. Buning uchun yuqoridagi funksiyalarda f ifodaning o’rniga berilgan integralni yozish kifoya. Misollar keltiramiz:

>restart;

>int(exp(sin(x)),x);

>taylor(%,x=0,4);

_esin( x) _dx





x  ¹ x²  ¹ x³  O( x⁵ )

2 6

>convert(%,polynom);

x  ¹ x²  ¹ x³
2 6

>taylor(int(sin(x)/x,x),x,8);

>convert(%,polynom);
1
x  ₁₈
x³ 
1

600
x⁵ 

1

35280
x⁷  O( x⁸ )

1
x  ₁₈
x³ 
1

600
x⁵ 

1 ₇

35280 ^x

>series(int(GAMMA(x),x),x=0,3);

ln( x )   x  ^_ ¹

1

2 2 2 3
   _ x  O( x )

>int(exp(x^3), x );


^_ 24
₄ _

( 1/3 ) _ 1

3  

1 ( 2/3 ) ^ 2 x ( -1)^{( 1/3 )}  3
x ( -1) _{ 3}, x _{ }

 ( -1) _
   _

³  ^{3  2  ( x3 )}( 1/3 )


_{( x3 )}( 1/3 ) _






_  ₃  

>series(%, x=0);

x  ¹ x⁴  O( x⁷ ) 4

>series(x^3/(x^4+4*x-5),x=infinity);

1 _ 4 _{
x x}4
5 _{ O} 1 _

 
_x5 _{ x}7 _

Quyidagi parametrga bog’liq xosmas integralga gamma funksiya deyiladi
+∞
Γ⁽𝑥⁾ = ∫ 𝑒^−𝑡 𝑡^𝑥−1 𝑑𝑡, 𝑥 > 0.
0
Maple dasturida gamma funksiyaning qiymatlarini aniqlash uchun GAMMA(x)
buyrug’idan foydalaniladi. Quyida gamma funkiyaga oid misollar keltirilgan:

• GAMMA(1/2);

• 
  GAMMA(1);
  

1

• series(int(GAMMA(x),x),x=0,3);

ln( x )   x  ^_ ¹
²  ¹ ^{2 }_ x²  _^  ¹ ( 3 )  ¹
²   ¹
^{3 }_ x³  O( x⁴ )

^_ 24
₄ _
 ₉
36 18 ^_

bunda 𝗒 −Eyler o’zgarmasi bo’lib, u quyidagiga teng:

1 1
γ = lim (1 + +
^𝑛→∞ 2 3
1
+ ⋯ +
𝑛
− ln 𝑛) = 0,5772156649 …,


ζ(x) – Rimanning dzeta funksiyasi : ζ(x)= _ ¹ , x  1 . Dzeta funksiyaning qiymatlarini
n1 ^nx
Maple muhitida Zeta(x) buyrug’i orqali aniqlanadi.

• Zeta(4);

• 
  Zeta(2);
  

• evalf(Zeta(3));

1 _4
90


1 _2
6

1.20205690

plot funksiyasida uzilishga ega bo’lgan funksiyalarning grafiklarini chizishda plot(…,discont=true) buyrug’idan foydalaniladi. color parametrida funksiya grafigini tasvirlovchi chiziqning rangi ko’rsatiladi, thickness parametrida esa grafikni tasvirlovchi chiziqning qalinligi ko’rsatiladi.

• plot(GAMMA(x),x=-5..5,y=-10..10,

thickness=2,discont=true,color=blue);

1 – 4 – mavzularni takrorlashga doir savol va topshiriqlar

1. 
   Maple muhitida berilgan integralni hisoblashdan asosiy maqsad nima?
   
2. 
   int va Int funksiyalarning vazifasini va ularning farqini ayting. Misollar bilan.
   
3. 
   Maple muhitida bo’laklab integrallash buyrug’ini ayting. Misollar bilan.
   
4. 
   Int(f(x),x); buyrug’idan so’ng, ekranda qanday natija hosil bo’ladi?
   
5. 
   Integralda o’zgaruvchini almashtirishdan maqsad nima?
   
6. 
   Maple muhitida integralda o’zgaruvchini almashtirib hisoblash buyrug’ini ayting.
   
7. 
   changevar(s,f,u); buyrug’ida s, f va u parametrlar vazifasini ayting va buyruqdan so’ng ekranda qanday natija hosil bo’ladi.
   
8. 
   Integralni hisoblashda value funksiyasi qanday vazifani bajaradi. Misollar bilan.
   
9. 
   Bo’laklab integrallash formulasini va uni Maple dasturida hisoblash funksiyasini ayting. Misollar bilan.
   
10. 
    Aniqmas integralni darajali qator ko’rinishda aniqlashdan maqsad nima?
    
11. 
    Maple muhitida aniqmas integralni darajali qator ko’rinishda hisoblash buyrug’ini ayting. Misollar bilan.
    
12. 
    1 – 4 – mavzularda keltirib o’tilgan funksiyalar amal bajarishi uchun bu funksiyalarga qadar yana qanday buyruq berib o’tilgan bo’lishi kerak?
    
13. 
    Maple muhitida bir vaqtning o’zida bir nechta funksiyalarning integrallarini topish uchun nima qilish kerak?
    
14. 
    Quyida keltirilgan funksiyalarning boshlang’ich funksiyalarini Maple muhitida hisoblang.
    

1	𝑓^(𝑥^)= 𝑥 − 3	13	𝑓^(𝑥^)= 𝑥^2 ∙ 3^√𝑥^2 − 3𝑥 + 6
2	1 𝑓^(𝑥^)= 𝑥^2 − + sin 𝑥 2𝑥 − 9	14	𝑥 𝑓^(𝑥^)= + tg^(sin 𝑥^)cos 𝑥 (𝑥^4 − 4)^3
3	𝑓^(𝑥^)= ^(3 − 𝑥^2^)2	15	arcsin(ln 𝑥 − 4) 𝑓^(𝑥^)= 𝑥
4	𝑓^(𝑥^)= ^(1 − 𝑥^)(1 − 2𝑥^)(1 − 3𝑥^)	16	𝑓^(𝑥^)= 2^𝑥 sin 𝑥

5	^(1 − 𝑥^)3 𝑓^(𝑥^)= 3 ^𝑥√^𝑥	17	𝑓^(𝑥^)= 𝑎𝑥^3 + log3^(sin^−1 𝑥 + 3^)cos 𝑥
6	1 ^𝑓^(^𝑥^)^=^(1^− )^√^𝑥√^𝑥 𝑥2	18	1 𝑓^(𝑥^)= sin 𝑥
7	3 √^𝑥^−^2^√𝑥2^+^1 𝑓^(𝑥^)= 4 √^𝑥	19	1 𝑓^(𝑥^)= 1 − cos 𝑥
8	2𝑥+1 − 5𝑥−1 𝑓^(𝑥^)= 10^𝑥	20	ln(𝑥 + √1 + 𝑥^2) 𝑓^(𝑥^)= ^√ 1 + 𝑥^2
9	𝑓^(𝑥^)= ^(2^𝑥 + 3^𝑥^)2	21	𝑥 𝑓^(𝑥^)= ^√1 + 𝑥^2 + √^(1 + 𝑥^2^)3
10	1 𝑓^(𝑥^)= sin^2 (2𝑥 + ^π) 4	22	sin 𝑥 𝑓^(𝑥^)= √^cos^2𝑥
11	√𝑥^2 + 1 − √𝑥^2 − 1 𝑓^(𝑥^)= √1 − 𝑥^4	23	1 𝑓^(𝑥^)= 3 cosh^2 𝑥 √tanh^2 𝑥
12	1 𝑓^(𝑥^)= 1 + sin 𝑥	24	𝑓^(𝑥^)= 𝑥^2^(2 − 3𝑥^2^)2

15. 
    Maple dasturida 𝑓⁽𝑥⁾ funksiyaning (a,A) nuqtadan o’tuvchi boshlang’ich funksiyasini toping va uning grafigini chizing.
    

1	1 𝑓^(𝑥^)= ; ^(0,1^)^(𝑥^2 − 𝑥 + 1^)√𝑥^2 + 𝑥 + 1	5	𝑓^(𝑥^)= ch^3 𝑥 sh^8 𝑥 ; ^(0,8^)
2	^36 ^𝑥^+^√𝑥2^+√^𝑥 3π 𝑓^(𝑥^)= 3 ; (1, ) ^𝑥(1^+√^𝑥)^2	6	𝑒𝑥 + 𝑒3𝑥 𝑓^(𝑥^)= ; ^(0, π^) 1 − 𝑒^2𝑥 + 𝑒^4𝑥
3	1 𝑓^(𝑥^)= ; ^(𝑒^𝑒, 1^) 𝑥 ln 𝑥 ln^(ln 𝑥^)	7	𝑓^(𝑥^)= sin ^4𝑥; ^(π, 0^)
4	^sin√^𝑥 π2 𝑓^(𝑥^)= ; ( , 0) √^𝑥^9	8	1 𝑓^(𝑥^)= 𝑥 ln^(1 + 𝑥^2^); (0, ) 2

Ko’rsatma: 𝑓(𝑥) funksiyaning (a,A) nuqtadan o’tuvchi boshlang’ich funksiyasidagi o’zgarmas C miqdor C = A −F(a) ekanidan foydalaning. Bunda F’(x) = 𝑓(x). Maple dasturida esa ushbu dasturdan foydalanish mumkin:

•