Paper 4: Fundamentals of Business Mathematics & Statistic

3.68 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 142:

ex

(^) x(1 xlogx)dx,x 0.+ >
I e logx dxx^1
x

=^ +^

Let ex log x = u

= du = u x x

e. e .logx dx du^1

x

(^) + (^) =
(^)
= ex log x + c. or. x
e^1 logx dx du
x
(^) + (^) =
(^)
Example 143:
3x dx
(^) 2x 1− Let 2x – 1 = u
(^2) , 2 dx = 2 udu, dx = udu
Again 2x = 1 + u^2 , x =^12 (1 u , 3x 1 u+^2 ) =^32 (+^2 )
3 (1 u )u 3^232 3 3 u^3
I=2 u + du du u du u .= 2 + 2 =2 2 3+
3u u 3^31 3/2
= 2 2 2+ = 2x 1 (2x 1) c.− + 2 − +
Example 144:Evaluate : 3x 6x 11dx^23 +
Let 6x^3 + 11 = u^2 so that 18x^2 dx = 2udu
( )
2 1 2 1 u 1^33 3/2
I 3 u.udu u du= 18 = 3 =3 3 9 = 6x 11 c.+ +
Example 145:Evaluate :
x (^2) dx
x 2

+

(^) − Let x – 2 = u
(^2) , dx = 2 udu. Again x + 2 = 4 +u (^2).

( )

4 u^22

I= +u .2udu 2 4 u du= +

=8 du 2 u du 8u u 8 x 2 x 2 c. + 2 = + 32 3 = − +^23 ( − )3/2+

Example 146:Evaluate:

x

x

e dx

1 e

−

(^) + −
Let 1 + e–x = u, –e-xdx = du
x x
x
I e dx du logu log(1 e )
1 e u
− −
= (^) + − = − = − = − +