Paper 4: Fundamentals of Business Mathematics & Statistic

3.82 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 166: 01 2xdx 2 xdx.= 01

2. 
   

   (^) ab f x g x dx f x dx g x dx( )± ( ) = (^) ab ( ) ± (^) ab ( )
   Example 167: (^01) (x 2x dx x dx 2 xdx.^2 ± ) = 01 2 ± 01

   
   


3. 
   

   ( ) ( )
   b a

   
   

(^) af x = − (^) bf x dx
i.e., interchange of limits indicate the change of sign.
Example 168: ( ) ( )
2 1
1 x 1 dx+ = − 2 x 1 dx.+

4. ( ) ( ) ( )
   b c b

(^) af x dx f x dx f x dx= (^) a + (^) c where a < c < b.
Example 169:
2 1 2
0 xdx xdx xdx,= 0 + 1 as 0 < 1 < 2.

SOLVED EXAMPLES

Example 170: (i)

2

1 x dx (ii)

(^14)
0 x dx (iii)
9
4 x dx (iv)
5
3
dx
(^) x (v)
(^2) 2x
1 e dx.
(i)
2 2 2 2 2
(^11)
x dx x 2 1 4 1 2 1 3.
2 2 2 2 2 2 2

=^ = − = − = − =

(^)
(ii)
1 5 1
4
(^00)
x dx x 1 0 1 0.^1
5 5 5 5 5

=^ = − = − =

(^)
(iii)
(^12)
9
(^91) 3/2 (^9) 3/2 3/2
4 4
4
x dx x^2 x^2 .9^24
(^11333)
2
(^) + (^)
= = = −
(^) +
(^)

=^23 (3 22.3/2− 2.3/2)=^23 (3 2^3 −^3 )= 32 (27 8− )=2.19 383 3=.

(iv) 35 dxx = logx log5 log3 log. 35 = − =^53

(v) ( ) ( )

2 2x^2

2x 2.2 2.1 4 2

(^11)
e dx e^1 e e^1 e e.
2 2 2

=^ = − = −

(^)