Maple dasturi yordamida aniqmas va aniq integrallarni hisoblash

>middlesum(sin(cos(x)+1),x=-Pi..Pi);

¹ _ ³ _

  ¹ 

¹  

 ^_

₂  _{  }sin_ cos_{ 2 } i  _{2 } _  1 _

>evalf(%);

_ i  0 ^{      }_

4.019502992

>middlesum(sin(cos(x)+1),x=-Pi..Pi,12);

_{1 } 11 _

  ¹ 

¹  

 ^_

₆  _{  }sin_ cos_{ 6 } i  _{2 } _  1 _

>evalf(%);

_ i  0 ^{      }_

4.045690562
Yuqorida keltirilgan buyruqlarni leftbox, rightbox va middlebox tarzida bersak, berilgan funksiya grafigi bilan chegaralangan egri chiziqli trapetsiyada saqlanuvchi, uni qoplovchi va orasida saqlanuvchi to’g’ri to’rtburchaklar bilan to’ldiradi. Boshqacha aytganda bu buyruqlar quyi, yuqori va o’rta integral yig’indilarni grafik tarzda tasvirlaydi.
Yana shuni unutmaslik kerakki, bu funksiyalar amal bajarishi uchun unga qadar
with(student): buyrug’i berib o’tilgan bo’lishi kerak.

• with(student):

• 
  leftbox(ln(x),x=1..10,12);
  

• rightbox(ln(x),x=1..10,12);

• 
  middlebox(ln(x),x=1..10,12);
  

>leftbox(x*sin(x),x=-3..5,12);

>rightbox(x*(2-ln(x^2))^3,x=-5..7,9);

>rightbox(x/(x^2+x+1),x=-10..10,30,shading=yellow);

>middlebox(sin(cos(x)+1),x=-10..10,20);

>middlebox(arcsinh(x),x=-10..10,20,color=black, thickness=4,shading=tan);

Aniq integralda ham bo’laklab integrallash mumkin. Bunda ham o’sha intparts

funksiyasidan foydalaniladi. Misollar bilan tanishamiz:

>restart;

>with(student):

>intparts(Int(x^n*exp(x),x=-10..10),x^n);

10

>evalf(%);

10ⁿ

_e10
 ( -10)ⁿ
_e(-10)
 _

^-10
xⁿ n e^x
_x dx
10.

22026.4657910.ⁿ

 .00004539992976( -10.)ⁿ
 1. ^

^-10.
xⁿ n e^x
_x dx

>intparts(Int(ln(ln(x))*sin(x),x=exp(exp(5))..exp(exp(10))),ln(ln(x

)));

e


10 cos( e^{( e}10 ⁾ )  5 cos( e^{( e}5 ⁾ )  ^


( e¹⁰ )

• 
  cos( x ) _dx x ln( x )
  

>evalf(%);

( e⁵ )
e

.332606399310⁹⁴⁹⁷

>intparts(Int(arcsin(x)/x^2,x=0.2..0.8),arcsin(x));

.8

>evalf(%);

.152329418 ^ 
_
^.2
¹ dx

1.446955071

3. 
   Aniq integralda o’zgaruvchilarni almashtirish uchun ham changevar funksiyasidan foydalaniladi. Misollar keltiramiz:
   

>restart;

>with(student):

>changevar(x=sin(u), Int(sqrt(1-x^2), x=a...b), u);

arcsin( b )

_
^arcsin( a )
cos( u ) du

>changevar(t=(1+x^0.25)^(1/3),Int((1+x^0.25)^(1/3)/x^0.5,x=a..b),t)

;

( 1.  b


_

_{( 1/4 )} ( 1/3 )
)
12.00000000t³ ( 1.  t³ )1.000000000 dt

>simplify(%);

( 1.  a

_{( 1/4 )} ( 1/3 )
)




12. ^_

( 1.  b
_{( 1/4 )} ( 1/3 )
)
t³ ( 1.  t³ ) dt

( 1.  a

_{( 1/4 )} ( 1/3 )
)

>changevar(u=x-1,Int(x^2-1,x=-3..2),u);

1

_
^_ ( 1  u )²  1 du
-4

>simplify(%);

>evalf(%);
1

_
^_ 2 u  u² du
-4
6.666666667

Trapetsiya usuli. 𝑓⁽𝑥⁾ funksiya ^[𝑎; P^] segmentda berilgan va 𝑎 = 𝓍₀ < 𝓍₁ < 𝓍₂ <
⋯ < 𝓍_𝑛 = P bo’lsin. Bulardan va
P − 𝑎 = 𝒽, 𝑓⁽𝑎⁾ = 𝑦₀, 𝑓⁽𝑥₁⁾ = 𝑦₁, … , 𝑓⁽𝑥_𝑛−1⁾ = 𝑦_𝑛−1, 𝑓⁽P⁾ = 𝑦_𝑛
belgilashlardan foydalanib
𝒽 𝑦₀ + 𝑦_𝑛
𝑛 ( ₂ + 𝑦₁ + ⋯ + 𝑦_𝑛)
yig’indini tuzamiz. 𝑛 ni yetarlicha katta tanlash hisobiga bu yig’indini 𝑓⁽𝑥⁾ funksiyaning integraliga yaqinlashtirish mumkin. Shuning uchun
P
∫ 𝑓⁽𝑥⁾𝑑𝑥 ≈ ^𝒽 (^{𝑦0 + 𝑦𝑛} + 𝑦 + ⋯ + 𝑦 )

_{𝑛 2} 1 𝑛
𝑎
deb yozishga haqqimiz bor. Bu formulaga trapetsiyalar formulasi deyiladi.
Maple dasturida trapetsiyalar formulasi bilan integralni taqribiy hisoblash uchun quyidagi buyruqlardan foydalaniladi:
>trapezoid(f(x),x=a..b);
>trapezoid(f(x),x=a..b,n);
bunda f(x) – integrali hisoblanayotgan funksiya, a va b lar mos ravishda integralning quyi va yuqori chegaralari, n - ^[𝑎; P^] ning bo’linishlari soni. Eslatib o’tamizki, 1 – buyruqda Maple natijani n=3 uchun chiqaradi.

Bu buyruqlardan so’ng buyruqqa mos natijanng matematik ko’rinishi ekranga chiqadi. Uning son qiymatini aniqlash uchun esa evalf funksiyasidan foydalaniladi.
Bu funksiyalar amal bajarishi uchun unga qadar with(student): buyrug’i berib o’tilgan bo’lishi kerak.

>with(student):

>int(4/(1+x^2),x=0..1);


>trapezoid(4/(1+x^2),x=0..1,1000);

_{3 1 } 999 _


¹ _ _

 _{  } 4

_
1000 1000 _{ i  1}  1
i²  1 ^

>evalf(%);

^ 
1000000
_ _

>trapezoid(f(x),x=a..b);

3.141592487

1 _ 1



¹ _ _

_ 3 _

_ 1 1


_ _ _

_{2  4} b  ₄ a _{ } f( a )  2 _{ } f_ a  i _{ 4} b  ₄ a _  f( b ) _



• evalf(%);

^ _{ } i  1 ^{ }
_{ }

>trapezoid(f(x),x=a..b,n);

_
 n  1 _


^{i ( b  a )} _ _ _

₁ ( b  a ) _ f( a )  2 _{ } f_ a 
_{n }  f( b ) _

_{ } i  1 ^
2 n
^ 

Parabola usuli (Simpson formulasi). Endi ^[𝓍₀; 𝓍₁^], ^[𝓍₁; 𝓍₂^], … , ^[𝓍_𝑛−1; 𝓍_𝑛^]
kesmalarning o’rta nuqtalari

da 𝑓⁽𝑥⁾ funksiyaning qiymatlarini

^𝜉1_⁄2 ^{, 𝜉}3_⁄2 ^{, … , 𝜉}_𝑛−1/2

^𝑦1_⁄2 ^{= 𝑓 (𝜉}1_⁄2^{) , 𝑦}3_⁄2 ^{= 𝑓 ( 𝜉}3_⁄2^{) , … , 𝑦}_𝑛−1_⁄2 ^{= 𝑓 (𝜉}_𝑛−1_⁄2⁾
ham hisoblab, quyidagi yig’indini tuzamiz:
𝒽
_6𝑛 (⁽𝑦₀ + 𝑦_𝑛⁾ + 2⁽𝑦₁ + ⋯ + 𝑦_𝑛−1⁾ + 4(𝑦_1/2 + 𝑦_3/2 + ⋯ + 𝑦_𝑛−1/2))
Bu yig’indi 𝑛 ning yetarlicha katta qiymatlarida 𝑓⁽𝑥⁾ funksiyaning integraliga yaqin qiymatlarni qabul qiladi. Shunga asosan
P
𝒽
∫ 𝑓⁽𝑥⁾𝑑𝑥 ≈ _6𝑛 (⁽𝑦₀ + 𝑦_𝑛⁾ + 2⁽𝑦₁ + ⋯ + 𝑦_𝑛−1⁾ + 4(𝑦_1/2 + 𝑦_3/2 + ⋯ + 𝑦_𝑛−1/2))
𝑎
formula o’rinli. Bu formulaga parabolalar yoki Simpson formulasi deyiladi.
Maple dasturida Simpson formulasi orqali berilgan funksiyaning integralini taqribiy qiymatini topish uchun

>simpson(f(x),x=a..b);

>simpson(f(x),x=a..b,n);
buyruqlaridan foydalaniladi. Bu buyruqlardan so’ng buyruqqa mos natijaning matematik ko’rinishi ekranga chiqadi. Uning son qiymatini aniqlash uchun esa evalf funksiyasidan foydalaniladi. 1 – buyruq Simpson formulasini n=2 uchun hisoblaydi.

>restart;

>with(student):

>int(4/(1+x^2),x=0..1);



>simpson(4/(1+x^2),x=0..1,10);

_  _
^{1 2} _ ⁵ _


¹ _ ¹ _ ⁴ _ ¹ _ _

 4
5 15
^{2  } 15 ^{   4} 1 ^

_ i  1 _
 1 _{i } 1 _
 1 ^
_ i  1 _
i²  1 _

>evalf(%);

_ _
^_ 5 10^_
_ _
^ _ 25
_ _

>simpson(f(x),x=a..b);

>simpson(f(x),x=a..b,n);

3.141592613

bo’lib,
Aytaylik, 𝑓⁽𝑥, 𝑦⁾ funksiya M = ^{(𝑥, 𝑦⁾: 𝑎 ⩽ 𝑥 ⩽ 𝑏, 𝑦 ∈ D ⊂ R^} to’plamda aniqlangan

𝑏
𝐹⁽𝑦⁾ = ∫ 𝑓⁽𝑥, 𝑦⁾𝑑𝑥

𝑎

integral mavjud bo’lsa, 𝐹⁽𝑦⁾ funksiyaga 𝒚 parametrga bog’liq aniq integral deyiladi. Maple muhitida parametrga bog’liq aniq integrallarni hisoblash uchun ham int funksiyasidan foydalaniladi.

•