Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.83

Example 171:Evaluate :
2 2
1

x 2x 5dx.

x

+ +

2 2 2

(^11)
I x 2 dx^5 x 2x 5logx
x 2

=^ + +^ =^ + +^

(^)
22 1
=^2 +2.2 5log2+^ −^2 + +2 5 log1 (^)

( ) ( )

6 5 log 2^5 as log1 0^7 5 log 2.

2 2

= + −^ = = +

(^)
Example 172:Evaluate :
1
0
1 xdx.
1 x

−

(^) +
(^) 1 x1 x+− dx= (^) 1 x+^2 −1 dx 2 = (^) 1 xdx+ −^ dx 2 log 1 x x^ (^ )^
∴ =I 2log 1 x x 2 log2 1 2 log1 2 log2 1 (+ )− 10 =( − −) = −
Example 173:Evaluate :
2
1
1 2 3x dx
x
(^) + +
(^)
1
1 2 3x dx dx 2 x dx 3 xdx^2
x
(^) + + (^) = −
x 3x1/2^2 3x^2
=x 2+ 1/2 2+ =x 4 x+ + 2

∴ I =

2 2

1

x 4 x 3x 2 4 2 .4 1 4^3 3 3 8 2.

2 2 2 2

(^) + + (^) = (^) + + (^) − (^) + + (^) = +
(^)
Example 174:Evaluate :
(^1) 2 x
0 x e dx
x e dx x e 2xe dx x e 2 xe e2 x 2 x x 2 x (^) ( x x)
= ex (x^2 – 2x + 2), (integrating by parts)
I = ( ) ( ) ( )
x 2^10
(^) e x 2x 2 e 1 2.1 2 e 0 0 2 e 2.− + 0 = − + − − + = −