Page 250 - DECO403_MATHEMATICS_FOR_ECONOMISTS_HINDI

VED1
E\L-LOVELY-H\math14-1 IInd 6-8-11 IIIrd 24-1-12 IVth 21-4-12 VIth 10-9-12

vFkZ'kkfL=k;ksa dk xf.kr

uksV
1 A B C
eku yks - + +
x (1 − x )(1 + ) x x 1 − x 1 + x
∴ - A (1 − x 2 ) + Bx (1 + ) x + Cx (2 − ) x

lehdj.k (1) esa Øe'k% - 3. vkSj 2 j[kus ij
1 1
- 1,B = , C = −
2 2
dx
∴ ∫ x − x 3 - ∫    1 + 2(1 − 1 ) x − 2(1 + 1 ) x    dx
x

1 1
+
- log||x − log|1 − x | − log|1 + x | c . mÙkj
2 2
mnkgj.k 7- ∫ dx dk eku Kkr dhft,A
4 − x 2

∫ dx
∫ dx
gy % 4 − x 2 (2) − x 2
2

1 2 + x  dx 1 a + x 
- log + , c  ∫ = log + c 
2.2 2 − x  a − x 2 2a a − x   
2
1 2 + x
- log + . c mÙkj
4 2 − x

mnkgj.k 8- eku Kkr dhft,% ∫ 1 dx
( + x )(x 2 + b a 2 )

gy % lekdY; dks vkaf'kd fHkUu esa tksM+us ij
1 A + Bx + C
2
(x + bx + )( 2 a 2 )
x + b x + a 2
1 1  1 b − x 
gy djus ij 2 2 - 2 2   + 2 2 2 2  
(x + bx + ) ( a ) a + b  x + b (a + b ) (x + a ) 

∴ - - ∫ 1 2 2 dx
(x + b ) (x + a )
1 dx 1 b − x
- 2 ∫ + 2 ∫ dx
a + 2 b x + a a + 2 b x + 2 a 2
1 b dx 1 2x
- log|x + b |+ 2 ∫ − 2 ∫ dx
a + 2 b 2 a + 2 b x + 2 a 2 2(a + 2 b ) x + 2 a 2
1  b 1 x 1 2 
- 2  log|x + b |+ tan − − log|x + 2 a | +  c
a + b  a a 2 
2
 
1 x + b b 1 x
-  log + tan −  + . c mÙkj
a + b  2 x + b 2 a a
2
2
 