Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.87

10. (i)

1

0

dx.

(^) x x+ [Ans. 2 log 2] (ii)
a
(^022)
xdx.
(^) a x− [Ans. a]
Show that :
11.
log2 x
0 x
e dx log. 3
(^) e 1+ = 2

12. 
    

    ( ) ( )
    2
    0 x x 1 x 2 dx 0.− − =

    
    


13. 
    

1 /2

0

dx 3 2.

3 2x

= −

(^) −

14. (i)

1

0

dx (^4) 2.
(^) 1 x x+ − = 3
(ii)
3
0
x dx (^14).
(^) x 1 5x 1+ + + = 15

15. ( )

e

1 2

dx (^1).
(^) x 1 logx+ = 2

16. ( )

e

2 2

(^11) dx e (^2).
logx logx log 2
(^)
− = −
(^)
Evaluate :

17. ( )

1 x

0 2

xe dx.

(^) x 1+ [Ans.
e 1
2 − ]
18.
3 5
2 4
x dx.
(^) x 1− [Ans.
5 1 4log
2 4 3+ ]
[Hints :
2 2
2 2
x u,1 u du 1 1 1 du u log1 1 u 1
2 u 1 2 u 1 2 4 u 1

= =^ +^ = + −

(^) − (^) − (^) + & etc.]
Summation of a Series by Definite Integral :
From the definition of definite integral, we know
( ) ( )
b n
(^) a f x dx limh f a rh ,=h 0→ r 1^ = + where nh = b – a.
For a = 0 and b = 1, we find nh = 1 – 0 = 1 or h =^1 n