Paper 4: Fundamentals of Business Mathematics & Statistic

3.70 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

3.5.2 INTEGRATION BY RATIONALISATION

In some cases rationalisation is required to avoid the surd in the numerator or denominator before integration.

The idea will be clear from the following example.

Example 151:Integrate:

dx

(^) x 1 x 1+ − −
Now x 1 x 1 x 1 x 1 x 1 x 1+ −^1 − = + −^1 − × x 1 x 1+ ++ + −−
= (x 1) (x 1) 2x 1 x 1 1+ −+ + −− = ( x 1 x 1+ + − )
I ( x 1 x 1)dx^11 x 1dx^1 x 1dx
= 2 + + − = 2 + + 2 −
(^1) .(x 1)1/2 1 1 (x 1)1/2 1 (^1) (x 1)3/2 (^1) (x 1 c.)3/2
2 1 1 2 1 1 3 3
2 2

+ + − +

= + ⋅ = + + − +

+ +

SELF EXAMINATION QUSTIONS

1. (2X 3 D+ )^4 χ ( )

(^1) 2X 3 , c is to be added in all answers 5
10
(^) +
(^)

2. (2 3X D− )^6 χ ( )
   Ans. 2x 3^17
   21

− −

3. 2x 5dx+ ( )
   Ans. 2x 5^1 3/2
   3

(^) +
(^)

4. 3 3x 4dx+ ( )
   Ans. 3x 5^1 4/3
   4

(^) +

5. (3x 4+ )5/3 ( )
   Ans. 3x 4^1 8/3
   8

(^) +
(^)

6. x x 1dx^2 + ( )
   Ans. x 1^12 3/2
   3

(^) +
(^)

7. (2x 5 x 5xdx+ )^2 + ( )
   Ans. x 5x^22 3/2
   3

(^) +

8. 2 2

xdx

(^) x a−
(^) Ans. x a^2 −^2

9.

3

2

x dx

(^) x 1+ ( )
Ans. x 1^12 3/2 x 1^2
3
(^) + − +
10. ( (^2) )^2
6xdx
(^) 4 x− 2
Ans.^1
4 3x
(^) − (^)

11. (3x 5x 7 6x 5 dx^2 − + )m( − )

( )

3x 5x 7^2 m 1

Ans. 3

(^) − + +