Integrallash
Kasr ratsional funksiyalarni sodda kasrlarga keltirish
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bo'laklab integrallash
- Bu sahifa navigatsiya:
- 2-teorema.
- 3-teorema.
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4. Kasr ratsional funksiyalarni sodda kasrlarga keltirish. Agar bizga biror ) ( ) (
Q x P n n (1) ko’rinishdagi ratsional kasr berilgan bo’lsa,
bu kasrni to’g’ri kasr deb ya'ni n
Buyerda Q m (x) ko’phad ildizlarining haqiqiy, kompleks va karrali bo’lishlari katta ahamiyatga ega.
aytiladiki bu son berilgan ko’phadni -Q m (a)=0 nolga aylantiradi. Masalan x 3 + x
2 -2x-8 ko’phadning x=2 ildizi bo’ladi
2
+ 2 2 - 2 2 - 8 = 0 =>0 = 0
Bizga algebradan ma'lumki, agar a,b,c,...,d sonlar m-darajali Q m (x) ko’phadning ildizlari bo’lsa, u holda bu ko’phadni
Q m (x)=a
0 (x-a)(x-b)(x-c)...(x-d) (2) ko’rinishda yozish mumkin. Agar Q m (x) ko’pqadning ildizlari takrorlanadigan ya'ni karrali bo’lsa
x=a ko’phadning k 1 karrali ildizi
x=b ko’phadning k 2 karrali ildizi
x=d ko’phadning k s karrali ildizi bo’lsa, u holda Q m (x) ko’phadni quyidagicha yozish mumkin: 8 3 2 1 ) ...( ) ( ) ( ) ( ) ( 0 k k k k m d x c x b x a x a x Q (3) (k
1 +k 2 +…+k 8 =m) Agar Q m (x) ning ildizi takrorlanmaydigan kompleks son bo’lsa, u holda: Q m (x)=(x 2 +p 1 x+q
1 ) (x
2 +p 2 x+q 2 )...(x 2 +p k x+q k ) (4) ko’rinishda bo’ladi. Agar Q
m (x) ko’phadning ildizlari kompleks karrali bo’lsa, bu holda ko’phadni quyidagicha ifodalash mumkin: Q m (x)=(x 2 +p 1 x+q
1 ) t 1 (x
2 +p 2 x+q 2 ) t 2 ...(x 2 +p k x+q k ) t r
(5) (t 1 +t 2 +…+t r =m)
1-teorema. Agar (1) to’g’ri ratsional kasr maxrajidagi Q m (x) ko’phad m ta haqiqiy har xil a,b,c,...,d ildizlarga ega bo’lib, (2) ko’rinishda bo’lsa, u holda (1) kasr I-tipdagi eng sodda kasrlarga ajraladi: d x D c x C b x B a x A x Q x P m n ...
) ( ) ( (6)
A,B,C,...,D lar noma'lum o’zgarmas koeffisiyentlar.
m (x) ko’phad ildizlari haqiqiy karrali bo’lib, (3) ko’rinishda bo’lsa, u holda (1) kasr I, II -tipdagi eng sodda kasrlarga ajraladi:
x D d x D d x D d x D b x B b x B b x B b x B a x A a x A a x A a x A x Q x P k k k k k k k k k k k k m n 8 8 8 8 2 2 2 2 1 1 1 1 ...
) ( ) ( ) ( ..... .......... .......... .......... .......... .......... .......... .......... .......... ...
) ( ) ( ) ( ... ) ( ) ( ) ( ) ( ) ( 2 3 1 2 1 2 3 1 2 1 2 3 1 2 1 (7) 3-teorema. Agar (J) to’g’ri kasr maxrajidagi Q m (x) ko’phad ildizlari takrorlanmaydigan kompleks sonlar bo’lib, (4) ko’rinishda bo’lsa, u holda (1) kasr III -tipdagi eng sodda kasrlarga ajraladi: k k k k m n q x p x N x M q x p x N x M q x p x N x M x Q x P 2 2 2 2 2 2 1 1 2 1 1 ) ( ) (
(8) M 1 ,N l M 2 ,N 2 , ... M k ,N k lar noma'lum o’zgarmas koeffisiyentlar. 4-teorema. Agar (1) to’g’ri kasr maxrajidagi Q m (x) ko’phad ildizlari kompleks karrali bo’lib, (5) ko’rinishda bo’lsa, u hoJda (1) kasr III va IV -tipdagi eng sodda kasriarga ajraladi: ) ( ... ) ( ) ( ...
.......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ) ( ... ) ( ) ( ... ) ( ) ( ) ( ) ( 2 1 2 2 2 2 1 1 2 2 2 1 2 2 2 2 2 2 2 2 1 1 1 1 2 1 1 1 2 2 2 1 1 2 1 1 2 2 2 2 2 2 1 1 1 1 r r t t t r r t r r t t t t t t t t m n q x p x B x A q x p x B x A q x p x B x A q x p x F x E q x p x F x E q x p x F x E q x p x N x M q x p x N x M q x p x N x M x Q x P r r (9)
M 1, N 1 , ... ,A
t ,B,
t lar noma'lura o’zgarmas koeffisiyentlar.
Shunday qilib (1) to’g’ri kasr raaxrajidagi Q m (x) ko’phadning ildizi haqiqiy har xil, haqiqiy karrali, kompleks va kompleks karrali bo’lsa, u holda (1) kasrni I,II,III,IV -tiplardagi eng sodda kasrlardan tashkil topgan (6),(7),(8),(9) ko’rinishdagi eng sodda kasrlarga ajratish murakin ekan.
(6),(7),(8),(9) ifodalardagi A I B I ,…,
D I , M I , N I
o’zgarmas noma'lumlarni topish uchun (6)-(9) tengliklarni (ayniyatlarni) umumiy maxrajga keltirib, chap va o’ng tomonlarida bir xil darajali ko’phad hosil qilamiz. So’ngra suratlaridagi chap va o’ng tomonidagi bir xil darajali x larning oldidagi koeffisiyentlarni tenglashtirib , yuqoridagi nomalum o’zgarmas A I
I ,…,
D I , M I , N I
koeffisiyentlarga nisbatan algebraik tenglamalar sistemasini hosil qilamiz va uni yechib noma'lum koeffisiyentlar aniqlanadi. So’ngra topilgan noma'lumlar o’rinlariga qo’yilib (6),(7),(8),(9) kasrlar bizga ma'lum bo’lgan eng sodda kasrlar kabi integrallananadi.
x x x x x 4 5 2 20 25 2 3 2 integrallang.
) 1 )( 4 ( ) 4 5 ( 4 5 2 2 3 x x x x x x x x x
1 4 ) 1 )( 4 ( 2 20 25 4 5 2 20 25 2 2 3 2 x D x B x A x x x x x x x x x x umumiy maxrajga keltirsak ) 1
4 ( ) 4 ( ) 1 ( ) 1 )( 4 ( ) 1 )( 4 ( 2 20 25 2
x x x Dx x Bx x x A x x x x x suratlarini tenglashtirib quyidagini hosil qilamiz: 25x
2 - 20x + 2 = A(x - 4)(x -1) + Bx(x -1) + Dx(x - 4)
25x
2 -20x + 2 = Ax 2 -Ax-4Ax + 4A + Bx 2 -Bx + Dx
2 -4Dx
O’ng va chap tomonidagi bir xil darajali x ning oldidagi koeffisiyentlarini tenglashtirib, quyidagi algebraik tenglamalar sistemasini hosil qilamiz:
3 7 ; 6 161 ; 2 1 4A 2 :
4D - B - 4A - -A 20 - : 25 : 0 2 D B A x x D B A x
) 1 ( 3 7 ) 4 ( 6 161 2 1 ) 1 )( 4 ( 2 20 25 2
x x x x x x x endi integrallasak C x x x x dx x dx dx x dx dx x x x x x 1 ln 3 7 4 ln 6 161 ln 2 1 1 3 7 4 6 161 2 1 ) 1 )( 4 ( 2 20 25 2
2-misol.
x x x x 2 3 2 3 ) 2 ( 4 2
2 ) 2 ( ) 2 ( 4 2 2 2 3 2 3 2 3 x F x E x D x B x A x x x x
2 3 3 3 2 2 2 2 2 3 2 3 ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( 4 2 x x x Fx Ex x Dx x Bx x A x x x x
x 3 -2x
2 +4 = Ax
2 -4Ax + 4A + Bx 3 -4Bx
2 +4Bx + Dx 4 -4Dx
3 + 4Dx
2 +Ex
3 + Fx
4 -2Fx
3
4 1 ; 2 1 , 4 1 , 1 , 1 4A 4
: 4B 4A 0
: 4D 4B - A 2 -
: B 2F - E -4D 1
: F D 0
: 0 2 3 4
E D B A x x x x x
) 2 ( 2 1 2 ln 4 1 ln 4 1 1 2 1 2 4 1 ) 2 ( 2 1 4 1 ) 2 ( 4 2 3 2 2 3 2 3 2 3 x x x x x x dx x dx x dx x dx x dx dx x x x x
) 5 2 )( 1 ( 2 2 x x x xdx integrallang. X 2 +1=0 i x i x 1
;
1 2 1 i x i x x x 2 1 ;
2 1
0 5 2 4 3 2
5 2 1 ) 5 2 )( 1 ( 2 2 2 2 1 1 2 2 x x N x M x N x M x x x x
) 5 2 )( 1 ( 2 2 x x x x = ) 5 2 )( 1 ( ) 1 )( ( ) 5 2 )( ( 2 2 2 2 2 2 1 1 x x x x N x M x x N x M
x=M 1 x 3 +2M 1 x 2 +5M
1 x+N
1 x 2 +2N 1 x+5N 1 +M 2 x 3 +N 2 x 2 +M 2 x+N 2 2 1 ; 10 1 ; 5 1 ; 5 1 5N 0
: 2 5M 1
: 2M 0 : M 0
: 2 1 2 1 2 1 0 2 1 1 2 1 1 2 2 1 3 N N M M N x M N x N N x M x
) 5 2 ( 10 3 2 2 ) 1 ( 10 1 2 ) 5 2 ( 10 5 2 ) 1 ( 10 1 2 ) 5 2 )( 1 ( 2 2 2 2 2 2
x x x x x x x x x x x x x
4 ) 1 ( 3 10 1 5 2 ) 2 2 ( 10 1 1 10 1 1 2 10 1 ) 5 2 )( 1 ( 2 2 2 2 2 2 x dx x x dx x x dx x xdx x x x xdx
x arctg x x arctgx x 2 1 20 3 5 2 ln 10 1 10 1 1 ln 10 1 2 2
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