FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.87
- (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
( ) ( )
2
0 x x 1 x 2 dx 0.− − =
1 /2
0
dx 3 2.
3 2x
= −
(^) −
- (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 :
- ( )
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