Maple dasturi yordamida aniqmas va aniq integrallarni hisoblash

int(f(x),x=a..infinity); buyrug’i qanday vazifani bajaradi?

Maple dasturida funksiyaning aniq integralini qaysi buyruqlar orqali hisoblanadi? Misollar bilan.

Maple dasturida funksiyaning xosmas integralini hisoblash buyrug’ini ayting? Misollar bilan.

Maple dasturida xosmas integrallarda bo’laklab integrallash va o’zgaruvchini almashtirish buyruqlarini ayting. Misolar bilan.

Quyidagi integrallarini hisoblang.

1	4 𝑑𝑥 ∫ ^1^+√^2𝑥^+ 1 0	6	2 ∫ 𝑓^(𝑥^)𝑑𝑥 , agar 𝑓^(𝑥^)= {^𝑥2^, 0^⩽^𝑥^⩽^1 𝑥, 1 < 𝑥 ⩽ 2 0
2	π ∫ 𝑥^3 sin 𝑥 𝑑𝑥 0	7	2 ∫^|1 − 𝑥^|𝑑𝑥 0
3	1 ∫ 𝑒^𝑥 arcsin 𝑒^−𝑥 𝑑𝑥 0	8	𝑒 ∫ sin ln 𝑥 𝑑𝑥 1
4	sinh 2 1 ∫ 𝑑𝑥 √1 + 𝑥^2 sinh 1	9	2 3 ^∫√^𝑥^+^√^𝑥2^𝑑𝑥 −1

5

3 ∫ sign^(𝑥 − 𝑥^3^)𝑑𝑥 0

10

6 π𝑥 ∫^[𝑥^]sin 𝑑𝑥 6 0

12. 
    Ko’rsatilgan almashtirishni bajarib integrallarni hisoblang.
    

1	𝑒3 1 ∫ 𝑑𝑥, ln 𝑥 = 𝑡 𝑥 ln 𝑥 ln^(ln 𝑥^) 𝑒𝑒	6	2𝑎 𝑥2 ∫ 𝑑𝑥 , 𝑥 = 𝑎 ∙ sinh 𝑡 √𝑎^2 + 𝑥^2 𝑎
2	5 𝑑𝑥 ∫ 3 , 𝑥 = sin 𝑡 2 ^(1 − 𝑥^2^)^2	7	π 2 ∫ cos^5 𝑥 √sin 𝑥 𝑑𝑥, sin 𝑥 = 𝑡^2 π 3
3	3 𝑥 ∫ 𝑥^√𝑑𝑥, 𝑥 = sin^2 𝑡 4 − 𝑥 2	8	7 1 ∫ 𝑑𝑥, 𝑥 − 1 = 8 sin^2 𝑡 √^(𝑥 − 1^)(8 − 𝑥^) 2
4	𝑒−1 ^arctan√^𝑥 1 ^∫^∙^𝑑𝑥,^arctan√^𝑥^=^𝑡 √𝑥 1 + 𝑥 1	9	𝑒 𝑑𝑥 ∫ 𝑑𝑥, tanh 𝑥 = 𝑡^3 cosh^2 𝑥 ∙ ^3√tanh^2 𝑥 1
5	2 ∫ (1 + 𝑥 − ^1) 𝑒^𝑥+1 𝑑𝑥, 𝑥 + ^1= 𝑡 𝑥 𝑥 𝑥 1 2	10	1 1 + 𝑥^2 1 ∫ 𝑑𝑥, 𝑥 − = 𝑡 1 + 𝑥^4 𝑥 −1

13. 
    Quyidagi integrallarda bo’laklab integrallash formulasidagi berilgan 𝒖 funksiya orqali bo’laklab integrallang va uni hisoblang.
    

    1	1 5 5 ∫ 𝑥^5 𝑒^𝑥 𝑑𝑥, 𝑢 = 𝑒^𝑥 −1	5	ln 4 𝑥𝑒^𝑥 ∫ 𝑑𝑥, 𝑢 = 𝑥𝑒^𝑥 ^(𝑥 + 1^)2 ln 3
    2	√^3 𝑒arctg 𝑥 1 ∫ 3 𝑑𝑥, 𝑢 = (1 + 𝑥^2)^2^√1^+^𝑥^2 0	6	π 3 𝑥 ∫ 𝑑𝑥, 𝑢 = 𝑥 cos^2 𝑥 π 4
    3	1 ∫ 𝑒^√𝑥 𝑑𝑥, 𝑢 = 𝑒^√𝑥 0	7	5π 6 ln(sin 𝑥) ∫ 𝑑𝑥, 𝑢 = ln(sin 𝑥) sin^2 𝑥 π 2
    4	𝑒 ln 𝑥 ^2 ln^2 𝑥 ∫ ( ) 𝑑𝑥, 𝑢 = 𝑥 𝑥 1	8	1 ∫ arcsin 𝑥 𝑑𝑥 , 𝑢 = arcsin 𝑥 0

14. 
    Quyidagi integrallarni 𝜀 aniqlikkacha taqribiy hisoblang.
    

1

1 ∫ sin 𝑥^2 𝑑𝑥, 𝜀 = 0.001 0

5

1 4 ∫ 𝑒^−𝑥2 𝑑𝑥, 𝜀 = 0.0001 0

2	1 2 ∫ cos 𝑥^2 𝑑𝑥, 𝜀 = 0.0001 0	6	π ∫ ^sin^𝑥𝑑𝑥, 𝜀 = 10^−8 𝑥7 π 2
3	1 16 ^∫^4√^𝑥^𝑒^−^𝑥2 ^𝑑^𝑥^, 𝜀^=^0^.^001 0	7	2 arctg 𝑥 ∫ 𝑑𝑥, 𝜀 = 10^−6 𝑥 1
4	2 ln^(1 + 𝑥^) ∫ 𝑑𝑥, 𝜀 = 0.00001 𝑥 1	8	π sin 𝑒^𝑥 ∫ 𝑑𝑥, 𝜀 = 10^−5 𝑒𝑥2 𝑒

15. 
    Quyidagi xosmas integrallarni hisoblang.
    

1	+∞ 𝑑𝑥 ∫ (1 + 𝑥 + 𝑥^2)^2 −∞	6	+∞ 𝑥𝑒arctg 𝑥 ∫ 𝑑𝑥 ^(1 + 𝑥^2^)√1 + 𝑥^2 0
2	+∞ 𝑥 ln 𝑥 ∫ (1 + 𝑥^2)^2 0	7	+∞ arctg^(1 − 𝑥^) ∫ 𝑑𝑥 ^3√^(𝑥 − 1^)4 0
3	1 𝑑𝑥 ∫ √1 − 𝑥^2 −1	8	+∞ tg 𝑥 ∫ 𝑑𝑥 𝑥 0
4	1 ∫ ln 𝑥 𝑑𝑥 0	9	+∞ ∫ sin^(𝑥^2^)𝑑𝑥 −∞
5	π 2 ∫ ln^(sin 𝑥^)𝑑𝑥 0	10	π 2 ∫ sin 𝑥 ln^(sin 𝑥^)𝑑𝑥 0

16. 
    Quyidagi parametrga bog’liq integrallarda parametr bo’yicha olingan hosilasini toping.
    

1	cos 𝑥 𝐹^(𝑡^)= ∫ cos^(π𝑡^3^)𝑑𝑡, 𝐹^′^(𝑡^)−? sin 𝑥	5	ln^(𝑥+𝑦^) 𝐹^(𝑥^)= ∫ 𝑥 + 𝑦Γ^(𝑦 + 𝑥^)𝑑𝑦, 𝐹^′^(𝑥^)−? ln^(𝑥−𝑦^)
2	𝑥2 𝐹^(𝑥^)= ∫ 𝑒^−^𝑥^𝑦2 𝑑𝑦, 𝐹^′^(𝑥^)−? 𝑥	6	𝑦2 sin^(𝑥𝑦^) 𝐹^(𝑦^)= ∫ 𝑑𝑥, 𝐹^′^(𝑦^)−? 𝑥 𝑥+𝑦
3	cos α 𝐹^(α^)= ∫ 𝑒^α^√^1^−^𝑥^2𝑑𝑥, 𝐹^′^(α^)−? sin α	7	𝑦^2cos^(𝑥+𝑦^) 𝐹^(𝑥^)= ∫ tg ^sin^(^𝑥^𝑦^)) 𝑑𝑦, 𝐹^′^(𝑥^)−? ( 𝑦 𝑥− 𝑦 ln^(𝑥𝑦^)

4

𝑥−2 𝐹^(𝑥^)= ∫ sin^(𝑦 − 𝑥^)𝑑𝑦, 𝐹^′^(𝑥^)−? 𝑥

8

𝑎𝑟𝑐𝑡𝑔^(𝑥−𝑦^) 𝐹^(𝑥^)= ∫ ln^(𝑦𝑥^(𝑥 + 𝑦^)^2^)𝑑𝑥, 𝐹^′^(𝑥^)−? arcsin 𝑦

1. 
   Biror chegaralangan Ω to’plamda aniqlangan 𝑓⁽𝑥, 𝑦⁾ funksiyaning ikki karrali integrali deganda, quyidagi songa aytiladi:
   

∬ 𝑓⁽𝑥, 𝑦⁾𝑑Ω = lim
max^|𝛥𝑥_𝑖^|→0
∑ ∑ 𝑓⁽𝑥_𝑖, 𝑦_𝑖⁾Δ𝑥_𝑖Δ𝑦_𝑖

^Ω max^|𝛥𝑦_𝑖^|→0 ^{𝑖 𝑗}
bunda Δ𝑥_𝑖 = 𝑥_𝑖+1 − 𝑥_𝑖, Δ𝑦_𝑗 = 𝑦_𝑗+1 − 𝑦_𝑗 va (𝑥_𝑖, 𝑦_𝑗) ∈ Ω . Agar Ω soha
𝑎 ⩽ 𝑥 ⩽ P, 𝓎₁⁽𝑥⁾ ⩽ 𝑦 ⩽ 𝓎₂⁽𝑥⁾ bunda 𝓎₁⁽𝑥⁾, 𝓎₂⁽𝑥⁾ ∈ C_[𝑎,P]
tengsizlik bilan berilgan bo’lsa, u holda yuqoridagi ikki karrali integral quyidagicha takroriy integralga keltirib hisoblanadi:
_{P 𝟐}⁽𝒙⁾ _{P 𝟐}⁽𝒙⁾
∬ 𝑓⁽𝑥, 𝑦⁾𝑑Ω = ∫ 𝑑𝑥 ∫ 𝑓⁽𝑥, 𝑦⁾ 𝑑𝑦 (= ∫ ( ∫ 𝑓⁽𝑥, 𝑦⁾ 𝑑𝑦) 𝑑𝑥)

Ω 𝑎
_𝟏⁽𝒙⁾
𝑎 _𝟏⁽𝒙⁾

Ikki karrali integralda o’zgaruvchilarni almashtirish:

Agar

𝑥 = 𝑥⁽𝑢, 𝑣⁾

𝑦 = 𝑦⁽𝑢, 𝑣⁾

𝗅
uzluksiz differensiallanuvchi funksiyalar sistemasi, O𝑥𝑦 tekislikdagi Ω sohani O𝑢𝑣
tekislikdagi Ω^′sohaga bir qiymatli akslantirsa va yakobiani
𝐷⁽𝑥, 𝑦⁾ 𝑥_𝑢 𝑦_𝑢

bo’lsa,
^{𝐼 =} 𝐷⁽𝑢, 𝑣^{) = |𝑥𝑣 𝑦𝑣 | ≠ 0}

∬ 𝑓⁽𝑥, 𝑦⁾𝑑Ω = ∬ 𝑓(𝑥⁽𝑢, 𝑣⁾, 𝑦⁽𝑢, 𝑣⁾)^|𝐼^| 𝑑Ω′

Ω Ω′
formula o’rinli.
Maple dasturida ikki karrali integralni hisoblash uchun
>Doubleint(f(x,y),x,y);
>Doubleint(f(x,y),x=a..b,y=c..d);
buyruqlaridan foydalaniladi. 1 – buyruq

∫ (∫ 𝑓⁽𝑥, 𝑦⁾𝑑𝑥) 𝑑𝑦
ning analitik ko’rinishini, 2 – buyruqda a, b, c, d lar o’zgarmas yoki o’zgaruvchi miqdorlar bo’lib,
𝑑 P
∫ 𝑑𝑦 ∫ 𝑓⁽𝑥, 𝑦⁾ 𝑑𝑥
𝓪
ning analitik ko’rinishini chop etadi. Demak,
>Doubleint(f(x,y),y=c..d, x=a..b);

buyrug’i
P 𝑑

∫ 𝑑𝑥 ∫ 𝑓⁽𝑥, 𝑦⁾ 𝑑𝑦
𝓪
ning matematik ifodasini chop etadi. Ularning qiymatini aniqlash uchun esa value
funksiyasidan foydalaniladi. Misollar bilan tanishamiz:

>restart;

>with(student):

>Doubleint(f(x,y),x,y);

^_^_f( x, y ) dx dy

>Doubleint(f(x,y),x=a..b,y=c..d);

d b

>Doubleint(x-y,x,y);

>value(%);
^_ ^_ f( x, y ) dx dy

 
c a
^_^_x  y dx dy
¹ x² y  ¹ y² x

2 2

>Doubleint(x+sin(x*y),x=1..2,y=1..3);

3 2
^_ x  sin( y x ) dx dy

>value(%);

₁ ₁

Ci( 3 )  3  Ci( 6 )  Ci( 1 )  Ci( 2 )

>Doubleint(x*y,y=x..x+1,x=2..3);

3
_ _
x  1
y x dy dx

>value(%);

₂ _x


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