x2 +J
5. Eva luat e : f ( X + 1)2
dx
.., 2
x- + I (x + I) - 2 x dx
Solu tion . f (x + I )2
dx - f (x + 1)2
- f I .dx - 2f x
(x+ 1)2
dx
( x + 1) - I dx
- x-2f (x + 1) 2
- x-2f I
x+ l
dx+2 I (x + 1)- 2
dx
(x+ l)- 1
= x - 2 log I x + 1 1· + 2 •+c
-1
2
- x - x+ - 2 log Ix + I I+ c , whi ch is the reqd. s
1
Or
dx
Eva lua te :
Jx 3 + x2 + x + 1 ·
I 1
Sol utio n. 3 2 - 2
(x +l) (x+ l)
x + x + x +l
A Bx+ C
- x+ l + x 2 +I
I '
1 = A (x2 + I) + (B x + C) (x + 1)
1
Put ting x =- I, 1 = A Cl+ 1) ⇒ A=
2.
l 1
1 =A + C ⇒ J=-+C ⇒ C=-
Put ting X = 0, 22·
l
Com par ing coe ffic ient s of x , 0 =A + B ⇒ B
2 = - A= - - .
2
, I I
f.rl llll I I ),
., \ t ')
,
I I t ' I I I I 1 I I I ·' j I
~f
). dr
r
\
,II ,I\
'. I , I I
"' I
,I . ' ) I I
I
I
/ 1
I I
I ,; I I It IHII
I
X +<
) lll}t I \ f I I log
2
4
I - I
+ I /- 4
I
Ing ( .,
2
+I) I- - tan ·'- +r ,
lo Q / 1 2
2 .. 2
1·. · r 2 ~ 0 =:- r2 + I > 0 :::. I x2 + I I = x + 11
-=~~!a_n - ' (x l) dx
.{ I + x4
2
So lu tion. 1 c t 1= f 2 X tan - I ( X )
- - --4 - d x
1+ x
Put tan -· 1.:r2 = t
2x
-- dx = dt .
so chac I+x
4
,2
1=
f I dt = 2 +C
I
= 2 (ta n- 1x2) 2 + c.
52
ssi ve ly with re pla ce me nt from a well sh uf fie d pa ck of
7. Two ca rds ar c dr aw n succe
tion of nu mb er of jac ks .
ca rd s. Find the pr ob ab ili t_y distribu of jacks .
. Lcl ·x · be the ra ndom va riable , which is the number
Solution
Herc X rakes val ues 0. 1. 2.
1 ) )2
p =
4
and q = l - - =_ .
Here 52 13 13 13
12 I2 I44
X- = -
P (X = 0) = - 13 l 69
13
P (X = 1) = 2 (- '
13
X !?13.)= 16249
1 1 1
P (X = 2) = - X - = - .
13 13 · 169
5. Eva luat e : f ( X + 1)2
dx
.., 2
x- + I (x + I) - 2 x dx
Solu tion . f (x + I )2
dx - f (x + 1)2
- f I .dx - 2f x
(x+ 1)2
dx
( x + 1) - I dx
- x-2f (x + 1) 2
- x-2f I
x+ l
dx+2 I (x + 1)- 2
dx
(x+ l)- 1
= x - 2 log I x + 1 1· + 2 •+c
-1
2
- x - x+ - 2 log Ix + I I+ c , whi ch is the reqd. s
1
Or
dx
Eva lua te :
Jx 3 + x2 + x + 1 ·
I 1
Sol utio n. 3 2 - 2
(x +l) (x+ l)
x + x + x +l
A Bx+ C
- x+ l + x 2 +I
I '
1 = A (x2 + I) + (B x + C) (x + 1)
1
Put ting x =- I, 1 = A Cl+ 1) ⇒ A=
2.
l 1
1 =A + C ⇒ J=-+C ⇒ C=-
Put ting X = 0, 22·
l
Com par ing coe ffic ient s of x , 0 =A + B ⇒ B
2 = - A= - - .
2
, I I
f.rl llll I I ),
., \ t ')
,
I I t ' I I I I 1 I I I ·' j I
~f
). dr
r
\
,II ,I\
'. I , I I
"' I
,I . ' ) I I
I
I
/ 1
I I
I ,; I I It IHII
I
X +<
) lll}t I \ f I I log
2
4
I - I
+ I /- 4
I
Ing ( .,
2
+I) I- - tan ·'- +r ,
lo Q / 1 2
2 .. 2
1·. · r 2 ~ 0 =:- r2 + I > 0 :::. I x2 + I I = x + 11
-=~~!a_n - ' (x l) dx
.{ I + x4
2
So lu tion. 1 c t 1= f 2 X tan - I ( X )
- - --4 - d x
1+ x
Put tan -· 1.:r2 = t
2x
-- dx = dt .
so chac I+x
4
,2
1=
f I dt = 2 +C
I
= 2 (ta n- 1x2) 2 + c.
52
ssi ve ly with re pla ce me nt from a well sh uf fie d pa ck of
7. Two ca rds ar c dr aw n succe
tion of nu mb er of jac ks .
ca rd s. Find the pr ob ab ili t_y distribu of jacks .
. Lcl ·x · be the ra ndom va riable , which is the number
Solution
Herc X rakes val ues 0. 1. 2.
1 ) )2
p =
4
and q = l - - =_ .
Here 52 13 13 13
12 I2 I44
X- = -
P (X = 0) = - 13 l 69
13
P (X = 1) = 2 (- '
13
X !?13.)= 16249
1 1 1
P (X = 2) = - X - = - .
13 13 · 169
