Paper 4: Fundamentals of Business Mathematics & Statistic

3.86 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 178:

(^22) x 1/x
1 2
x 1e dx.
x
(^) − (^) +
(^)
Let
2
2 2
x^1 u or, 1^1 dx du or,x 1dx du
x x x

+ =^ −^ = − =

(^)
For x = 2, u = 2 +2 21 5= ; x = 1, u = 1 + 1 = 2
5/2 u u 5/2 5/2 2
∴ =I 2 e du e=^2 =e e .−
Example 179:Evaluate :
1
0
dx
(^) x 1 x+ +
dx dx( x 1 x) ( x 1 x)dx
x 1 x ( x 1 x)

= + − = + −

(^) + + (^) + +
(^2) (x 1)3/2 (^2) x3/2
= 3 + − 3

( ) ( ) ( )

3/2 3/2^1 2/3 3/2

0

I^2 x 1 x^2 2 1 1 (2 2)^24 2 1.

3 3 3 3

∴ =^ + −^ =^ − −^ = − = −

SELF EXAMINATION QUESTIONS

Find the value of :

1.

2 2

0 3

x dx.

(^) 1 x+ [Ans.

4

3 ] 2.

(^22)
0 x 4 x dx.− [Ans.

8

3 ]

3.

2

0 2

6 x 5 dx.

3x 5x 1

+

(^) + + [Ans. log 23] 4.
3
0 x 1dx.+ [Ans.

14

3 ]

5. 0 1/3 1 3x dx.− [Ans. 92 ] 6. 10 x 1 3x dx.^3 +^4 [Ans. 187 ]

7.

(^223)
1 x x 8dx.+ [Ans.

74

9 ]

8. (i) ( )

e^2

1

dx.

(^) x 1 logx+ [Ans. log 3] (ii)
2
e
dx.
(^) xlogx [Ans. log (log 2)]

9. ( )

e^2

1 2

dx.

(^) x 1 logx+ [Ans.

2

3 ]