Maple dasturi yordamida aniqmas va aniq integrallarni hisoblash

F:=unapply(int(f(x),x),x):

C:=A-F(a):

int(f(x),x)+C;

16. 
    Quyidagi bo’lakli funksiyalarning boshlang’ich funksiyasini Maple muhitida toping va uning grafigini chizing.
    

1	𝑓^(𝑥^)= ^|1 + 𝑥^|− ^|1 − 𝑥^|	10	sin 𝑥 , 𝑥 < 0 𝑓^(𝑥^)= {𝑥 + 1, 0 ⩽ 𝑥 ⩽ 1 𝑒^𝑥 , 𝑥 > 1
2	𝑓^(𝑥^)= ^|2𝑥 − 7^|+ ^|3 − 8𝑥^|	11	( ) 1 − 𝑥^2, ^|𝑥^|⩽ 1 ^𝑓^𝑥^=^{1 − ^|𝑥^|, ^|𝑥^|> 1
3	^|𝑥^|− ^|𝑥^2 − 4^| 𝑓^(𝑥^)= |𝑥 − 9|	12	max(𝑥^2 − 4, 𝑥) , 𝑥 < 0 𝑓^(𝑥^)= { min(𝑥^2 − 9, 𝑥) , 𝑥 ≥ 0
4	𝑓^(𝑥^)= min^(𝑥, 𝑥^3 − 𝑥^)	13	^|2𝑥 + 9^|𝑒^𝑥, 𝑥 < 0 𝑓^(𝑥^)= { 𝑥^2 − sin 𝑥 , 𝑥 ≥ 0
5	𝑓^(𝑥^)= max^(𝑥 − 5,7 − 𝑥^)	14	𝑓^(𝑥^)= sign𝑥 + ^|𝑥^|
6	𝑓^(𝑥^)= min^(|𝑥 − 6^|, 6^)	15	𝑥 − 9 𝑓^(𝑥^)= sign^(2𝑥 − 8^)+ | | 𝑥
7	𝑓^(𝑥^)= ^[𝑥^]max^(2𝑥 − 5, 7 − 5𝑥^)	16	𝑥 sin 𝑥 , 𝑥 < 0 𝑥𝑒^𝑥 , 0 ⩽ 𝑥 < 10 ^𝑓^(^𝑥^)^=^{ln 𝑥 + 12 , 10 < 𝑥 ⩽ 20 max^(𝑥^2 − 4,3𝑥^), 𝑥 > 20
8	𝑓^(𝑥^)= ^|[𝑥^]− 9^|	17	𝑥 + sign𝑥, 𝑥 ∈ ^[−10,2^) 𝑓^(𝑥^)= { ( ) 4 𝜒[2,8] 𝑥 𝑥 , 𝑥 ≥ 2
9	𝑓^(𝑥^)= 𝜒[−1,1]^(𝑥^)	18	𝑓^(𝑥^)= 𝜒[2,8]^(𝑥 − 9^)

Eslatma: A to’plamning xarakteristik funksiyasi quyidagicha ta’riflanadi:

𝜒_A⁽𝑥⁾ =

1, 𝑥 ∈ A

0, 𝑥 ∉ A

𝗅

17. 
    Quyidagi funksiyalarning boshlang’ich funksiyalarini darajali qator ko’rinishda dastlabki 5 – qo’shiluvchigacha aniqlang.
    

1	sin 2𝑥 𝑓^(𝑥^)= 𝑥	5	𝑥100 𝑓^(𝑥^)= ln 𝑥	9	𝑓^(𝑥^)= sin(𝑥)𝑒^𝑥
2	−𝑥2 𝑒 2 𝑓^(𝑥^)= √^2^π	6	𝑓^(𝑥^)= 𝑒^𝑥 ln^|𝑥^|	10	−𝑥2 𝑓^(𝑥^)= 𝑥𝑒 ^2
3	5 𝑓^(𝑥^)= ln 5𝑥	7	1 𝑓^(𝑥^)= √1 − 𝑘^2 sin^2 𝑥	11	𝑓^(𝑥^)= 𝑒^sin𝑥
4	𝑥 𝑓^(𝑥^)= cos 𝑥	8	1 𝑓^(𝑥^)= (1 + 2 sin^2 𝑥)^√1 − ^sin2 ^𝑥 4	12	𝑓^(𝑥^)= sin^(𝑐𝑜𝑠 𝑥^)

18. 
    Maple muhitida aniqmas integrallarni hisoblash funksiyalaridan foydalanib, quyidagi misollarni to’liq tahlil qiling.(Integralni uch xil usulda hisoblang(bevosita, bo’laklab, o’zgaruvchini almashtirib); ko’rsatilgan nuqtadan o’tuvchi boshlang’ich
    

funksiyasini topib, grafigini chizing; boshlang’ich funksiyani darajali qator ko’rinishda aniqlang).

1	ln 𝑥 𝑦 = 𝑥	7	3 √^𝑥^+ 2 𝑦 = 3 ^𝑥^+√^𝑥^+^2	13	𝑦 = 𝑥 ln 𝑥
2	𝑦 = sin 𝑥 + cos 𝑥	8	𝑦 = 𝑒^𝑥 + 𝑒^2𝑥 cos 𝑥	14	𝑦 = 𝑥ln 𝑥
3	𝑦 = tg 𝑥	9	𝑥2 𝑦 = 9 𝑥^4 + 3𝑥^3 + 𝑥^2 + 3𝑥 + 1 2	15	1 𝑦 = 3 ^(^1^+^4√^𝑥^)√^𝑥
4	𝑒ln 𝑥 𝑦 = 𝑥	10	1 ^𝑦^=^(1 + 𝑥^)(1 + 𝑥^2^)(1 + 𝑥^3^)	16	𝑦 = cos^5 𝑥
5	𝑥 𝑦 = 1 + 𝑥^2	11	𝑦 = sin^(cos 𝑥^)sin 𝑥	17	𝑦 = sin^4 𝑥 cos^5 2𝑥
6	𝑦 = sin 𝑥 ∙ ln(ctg𝑥)	12	1 𝑦 = ^1^+√^𝑥	18	2 𝑦 = sin 𝑥 − cos 𝑥

𝑓⁽𝑥⁾ funksiya ^[𝑎; P^] segmentda aniqlangan va 𝓪 = 𝑥_𝟎 < 𝑥_𝟏 < 𝑥_𝟐 < ⋯ < 𝑥_𝒏 = P bo’lsin. U holda 𝑓⁽𝑥⁾ funksiyaning ^[𝑎; P^] segmentdagi integrali deb quyidagi songa aytiladi:

P
∫ 𝑓⁽𝑥⁾𝑑𝑥 = lim
max^|∆𝓍_𝑖^|→0
𝑎
𝑛−1
∑ 𝑓⁽𝜉_𝑖⁾ ∆𝓍_𝑖
𝑖=0

bunda 𝑥_𝒊 ⩽ 𝝃_𝒊 ⩽ 𝑥_𝒊+𝟏 va ∆𝑥_𝒊 = 𝑥_𝒊+𝟏 − 𝑥_𝒊. Funksiyaning segmentdagi bu kabi aniqlangan integrali uning shu segmentdagi aniq integrali deb ham ataladi.
𝑓⁽𝑥⁾ funksiyaning aniq integraliga tegishli ayrim tushunchalarni keltirib o’tamiz:
1)
𝑛−1
𝖲 = ∑ 𝑚_𝑖 ∆𝓍_𝑖
𝑖=0
yig’indi 𝑓 funksiyaning quyi yoki o’ng integral yig’indisi deyiladi.
2)

𝑛−1
𝖲 = ∑ 𝑀_𝑖 ∆𝓍_𝑖
𝑖=0
yig’indi esa 𝑓 funksiyaning yuqori yoki chap integral yig’indisi deyiladi.
3)

𝑛−1
𝖲 = ∑ ℋ_𝑖 ∆𝓍_𝑖
𝑖=0
yig’indi 𝑓 funksiyaning o’rta integral yig’indisi deb ataladi. Bularda

𝑚_𝑖
= inf
𝓍_𝑖⩽𝑥⩽𝓍_𝑖+1
𝑓⁽𝑥⁾ , 𝑀_𝑖
= sup
𝓍_𝑖⩽𝑥⩽𝓍_𝑖+1
𝑓⁽𝑥⁾ ,
^𝓍𝑖 ^{+ 𝓍}𝑖+1
ℋ_𝑖 = 𝑓 ( ₂ )

Bunday olib qaralganda bu integral yig’indilar berilgan funksiyaning grafigi bilan chegaralangan egri chiziqli trapetsiyada saqlanuvchi, uni o’zida saqlovchi va orasida yotuvchi to’g’ri to’rtburchakchalar yuzalari yig’indisini ifodalaydi. Buni quyidagi chizmalarda ko’rish mumkin:

Maple dasturi yordamida bularni quyidagi funksiyalar orqali aniqlaymiz:

1. 
   int(f(x),x=a..b); buyrug’i f funksiyaning [a;b] dagi aniq integralini hisoblaydi.
   

>restart;

>int(x^3,x=-1..3);

>int(x*ln(x),x=1..t);

>int(mu(x),x=a..b);
20


¹ t² ln( t )  ¹ t²  ¹
2 4 4

>Int(sin(x),x=-Pi..Pi);

b


^_ ( x ) dx
a




_^ sin( x ) dx


>Int((1+x-1/x)*exp(x+1/x),x=1/2..2)=int((1+x- 1/x)*exp(x+1/x),x=1/2..2);

2
^{ } ^{x  1 }

 ^e




^1/2
^ ^{1  x }

¹  ^
_x _
x ^_{ dx } 3 _e( 5/2 )
2

>int(abs(x-2),x=0..2);

2

>Diff(Int(sin(x^2),x=a..b),a)=diff(int(sin(x^2),x=a..b),a);



b
^ sin( x² ) dx  sin( a² )
a ^
a
>Limit(Int(cos(t^2),t=0..x)/x,x=0)=limit(int(cos(t^2),t=0..x)/x,x=0);
x


^_ cos( t² ) dt

lim ⁰  1
x  0 ^x

2. 
   leftsum(f(x),x=a..b); leftsum(f(x),x=a..b,n); rightsum(f(x),x=a..b); yoki rightsum(f(x),x=a..b,n); middlesum(f(x),x=a..b); middlesum(f(x),x=a..b,n);
   

buyruqlari mos ravishda f funksiyaning [a;b] dagi yuqori, quyi va o’rta integral yig’indilarini hisoblaydi. Bunda n – to’g’ri to’rtburchaklar (yoki [a;b] kesmaning bo’linishlari soni) soni. Maple bu yig’indilarni kompleks ko’rinishini beradi. Uning haqiqiy qiymatini aniqlash uchun evalf funksiyasidan foydalaniladi.

>restart;

>with(student):

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

_ ³ _
2 _{ } ( 3  2 i ) sin( 3  2 i ) _

>evalf(%);

_ i  0


5.059324036

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

^{2 }_ 11  ²  



² _ _

_{3   }3  ₃ i _ sin_ 3  ₃ i _

>evalf(%);

_ i  0 ^
  _

2.384315849

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

 ^{4 3} 

>evalf(%);

3 _{ } ( 5  3 i ) ( 2  ln( ( 5  3 i )² ) ) _




_ i  1 _

-125.1073811

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

^{12 } ⁷
₁₂  
^{12 2  }3 ^

^ _ ^_ 5 


i ^_{ } 2  ln_ ^_ 5 


^{i }   ^

>evalf(%);

^{7 }_ i  1 ^
₇ _ _ _ _
7 _ _ _{ }

-55.30787680