A L g e b r a belgilar va belgilashlar
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Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 66 I N T E G R A L L A R 1. N'yuton-Leybnis formulasi: ( ) ( )
( ) ( ).
b b a a f x dx F x F b F a S = = = - ò 2. Egri chiziq bilan chegaralangan yuzalarni hisoblash: a) Egrichiziqli trapesiya yuzi: ( )
= ò ; b) agar 1 2
( ) 0
f x > > bo`lsa, u holda [ ] 1 2 ( ) ( ) ;
a S f x f x d x = - ò bo`ladi. 3. (
( ) ( ) 0
f x f x = > egri chiziq aylanganda hosil bo'lgan jism hajmi: 2 2
. b b a a V f x dx y dx p p = = ò ò 4. » : ( ), AB y f x a x b = £ £ yoyning uzunligi: 2 1 ( ) b a f x dx l ¢ = + ò . 5. » ( ), : ( ),
x x t AB y y t t a b = ì í = £ £ î
2 2
( ) x t y t dt l b a ¢ ¢ = + ò . 6. ( ) ( ) ( ) 0
f x f x = ³ , [ ]
, x a b Î egri chiziqni OX o`qi atrofida aylantirishdan hosil bo'lgan aylanish sirtining yuzini topish: 2 0 2 ( )
1 ( )
b f x f x d x S p ¢ = × + ò .
1. (
( ) , .
b b a a k f x d x k f x d x k c o n s t × = = ò ò 2. [ ] ( ) ( ) ( ) ( ) . b b b a a a f x g x d x f x d x g x d x + = + ò ò ò Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 67 3. ( ) ( ) ( )
( ) ( )
( ). b b b a a a f x d g x f x g x g x d f x = × - ò ò 4. ( ) 1 ( ) , 0, ` .
b a a f kx c dx k f kx c k c o zgarmas sonlar ¢ + = × + ¹ - ò 5. Agar [ ] ( ) ( ),
;
,
0 f x f x x a a a - =
Î - > bo`lsa, 0 ( )
2 ( )
a a a f x dx f x dx - = ò ò bo`ladi. 6. Agar [ ] ( ) ( ), ; , 0 f x f x x a a a - = -
Î - > bo`lsa, ( ) 0
a f x dx - = ò bo`ladi. 7. Agar [ ]
( ) 0,
; f x x a b ³ Î bo`lsa, ( )
0 b a f x d x ³ ò bo`ladi. 8. Agar
( )
0 ;
( ) 0
x c da f x c x b da f x < < ³
< bo`lsa, ( ) ( )
( ) b c a a a c f x d x f x d x f x d x = - ò ò ò bo`ladi. Aniqmas integral 1 1 1. . 2. .
m m dx ln ln x C sin x cosxdx sin x C x lnx n + = + × = + × + ò ò ( ) ( ) 2 2 2 1 1 3. 1 . 4.
1 . 1 1 1 x dx ln x x C x ln x dx x C x a a a a a + æ ö = - - + × = × - + ¹ - ç ÷ ç ÷ + + - è ø ò ò 2 2 2 2 2 5.
. 2 2
a x a x dx a x arcsin C a - = - + + ò ( ) 2 1 6. 1 . 2
x arctgx ln x C = ×
- × + + ò ( ) 7. 1 .
x x e dx x e C × = - × + ò ( ) 2 2 8. 2 2 . x x x e dx x x e C = - + × +
ò 2 2 1 1 9. 2 .
10. 2 . 2 4 2 4 x x sin xdx sin x C cos xdx sin x C = - + = + + ò ò 3 3 3 3 11.
. 12.
. 3 3 cos x sin x sin x dx cos x C cos x dx sinx C = -
+ + = - + ò ò 1 13.
. ln xdx x ln x ln x dx C a a a a - = × - + ò ò
A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 68 2 14. 1 . arcsin x dx x arcsin x x C = ×
+ - + ò 2 2 2 2 2 2 2 2 2 2 , 4 , 4 4 15. 1 2 4 , 4 . 4 2 4 cx b arctg C agar b ac ac b ac b dx a bx cx cx b b ac ln C agar b ac b ac cx b b ac + ì + < ï - - ï = í + + + -
- ï + > ï - + + - î ò 2 2 1 16.
2 2 , 0. dx ln cx b c a bx cx C c c a bx cx = + + + + + > + + ò 2 2 2 17.
4 cx b a bx cx dx a bx cx c + + + = + + - ò 2 2 2 4 2 2 . 8
a c ln c x b c a b x cx C c - - + + + + + 2 2 2 2 1 2 4 1 8. . 4 2 4 dx cx b b ac ln C a bx cx b a c cx b b a c - +
+ = + + - + - + +
+ ò 2 2 1 2 19. ,
0. 4
cx b arcsin C c c a bx cx b ac - = + > + - + ò 2 2 2 20. 4
b a bx cx dx a bx cx c - + - = + - + ò 2 2 2 4 2 . 8 4
ac cx b arcsin C c b ac + - + + + ( )( ) ( ) ( ) 21. . a x dx a x b x a b ln a x b x C b x + = + + + - + + + + + ò ( )( ) (
) 22. .
a x a x dx a x b x a b arcsin C b x a b - + = + + + + + + + ò ( )( ) ( ) 23. .
a x b x dx a x b x a b arcsin C b x a b + - = - + - - + + - + ò 2 4 . ,
. s h x d x ch x C ch x d x s h x C = + = + ò ò 25.
, .
lnchx C cthxdx lnshx C = + = + ò ò ( ) ( ) ( ) ( ) 26. ,
. 2 2 sin m n x sin m n x sin mx sin nx dx C m n m n m n + - × = -
+ + ¹ + - ò Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 69 ( ) ( ) ( ) ( ) 27. ,
. 2 2 sin m n x sin m n x cos mx cos nx dx C m n m n m n + - × = + + ¹ + - ò ( ) ( ) ( ) ( ) 28. , .
2 cos m n x cos m n x sin mx cos nx dx C m n m n m n + - × = -
- + ¹ + - ò ( ) 2 2 29. ( ) .
ax e sin nx dx e a sin nx n cos nx a n C × = × - ×
+ + ò ( ) 2 2 30. ( ) .
ax e cos nx dx e a sin nx n cos nx a n C × = × + ×
+ + ò 2 2 2 2 2 , , 2 31.
1 2 , . 2 a b x arctg tg C agar a b a b a b dx x b a tg a b a bcosx ln C agar a b x b a b a tg a b ì æ ö - × + > ï ç ÷ ç ÷ + - ï è ø ï = í
- × + + + ï + < ï - - × - + ï î ò 2 2 2 2 2 2 2 2 2 2 2 2 ,
, 32.
s 1 2 , . 2
a tg b arctg C agar a b a b a b dx x a b inx a tg b b a ln C agar a b x b a a tg b b a ì × + ï + > ï - - ïï = í + × + - - ï + < ï - ï × + + - ïî ò Download 0.8 Mb. Do'stlaringiz bilan baham: 
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