Paper 4: Fundamentals of Business Mathematics & Statistic

3.60 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

(iv) (a x) dx+^2 = (a 2ax x )dx a dx 2a xdx x dx^2 + +^2 =^2 + + 2

2 x x^232212

=a x 2a+ 2 3+ =a x ax x c.+ + 3 +

Example 119:Find the value of:

(i)

x x 4 x^5 dx

x

(^) − +
(ii) x x x 1 dx^2 ( − +)
Solution:
(i) (x x 4 x )dx x xdx 4 xdx 5^5 dx
x x
− + = − +
= x dx 4 x dx 5 x dx3/2 − 1/2 + −1/2

=

x3/2 1 x1/2 1 x1/2 1

3 4. 1 5. 1

2 1 2 1 2 1

+ + − +

− +

+ + − +

(^2) x 4. x 5.2x5/2 (^2) 3/2 1/2 (^2) x5/2 (^8) x 10 x c3/2
= 5 − 3 + = 5 − 3 + +
(ii) x x x 1 dx^2 ( − +) = (x x x dx x5/2−^3 +^2 ) = 5/2− x dx x dx^3 + 2
x5/2 1 x x 2^43 7/2 x x^43
(^5) 4 3 7x 4 3 c.
2 1

• = − + = − + +


• 
  

  Problem of algebraic functions:
  To integrate a fraction algebraic expression of which the numerator is a polynomial function and the
  denominator is a monomial (or binomial) function, simplify the expression first to partial fraction.
  Example 120:(integrate)
  (i)
  2x 3x 4^23
  +x + dx (ii) ( )
  x 22
  dx
  x

  
  

+

(iii)

2x 14x 24^2

−x 3− + dx

Solution:

(i) Same as

2x 3x 4^23

+x + dx

x^2 x^3 dx 2 dx

=2 dx 3 dx 4 (^) x + (^) x + (^) x =2 xdx 3 x dx 4 + + (^) x
x x^2323
=2 3. 4logx x x 4logx c.2 3+ + = + + +