Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.69

( )

x x x

x x

log 1^1 log e 1 loge log e 1 c.

e e

= −^ +^ = −^ +^ = − + +

(^)
Example 147:Evaluate : e e 1 dxx x( + )1/2
Let (ex + 1)1/2 = u, ex + 1 = u^2 , exdx = 2udu
( )
I 2 u du u^2 2 2^3 e 1 cx 3/2
= =3 3= + +
Example 148:Evaluate : ( )
2x
2
x.e dx
(^) 2x 1+
Let ( )
e2x
2x 1+ =u so that on differentiation

( ) ( )

2x 2x 2x

2 2

2(2x 1)e e 2dx du, 4xe dx du

2x 1 2x 1

+ − ⋅ = =

+ +

Now ( )

1 1 1 e^2 x

I du u= 4 =4 4 2x 1= + +c

Example 149:Evaluate :
2 x 3dx
4x 1

−

(^) −
let 4x – 1= u^2 , 4dx = 2 udu, again 2x 1 u ,2x 3 u 5=^12 (+^2 ) − = 21 (^2 − )

( )

1 u 5^212 1 u^3

I=2 2u− .udu= 4 u 5 du− =4 3^ −5u

(^)
1 (4x 1)3/2 5 4x 1 (^1) (4x 1)3/2 (^5) 4x 1 c
4 3 12 4
(^) −
= − − = − − − +
Example 150: x 3x 4^23 − dx
Let 3x^3 – 4 = u^2 , 9x^2 dx = 2 udu
( )
2 2 2 u 2^33 3/2
I u du .= 9 =9 3 27= 3x 4 c− +