Paper 4: Fundamentals of Business Mathematics & Statistic

3.66 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 133:

dx

(^) 2 5x+
Let 2 + 5x = u
5.dx = du
dx = 51 du
I = 5 u1 du
=^15 logu
=^15 log(2 5x) c+ +
Example 134: (^) ax bx c2ax b (^2) + ++ dx= dzz =logz c log(ax bx c) c.+ 1 =^2 + + + 1
Let ax^2 + bx + c = z, (2ax + b) dx = dz.
Note: Numerator is derivative of denominator.
Now (^) 2x 3x 4 (^24) +x b+ + dx log(2x 3x 4) c.=^2 + + +
(^232)
3 2
3x 4x 1 dx log(x 3x x 1) c.
x 2x x 1

+ + = + + − +

(^) + + −
Example 135: x 1 x dx^2 +^3
Let 1 + x^3 = u^2 (here u^2 is taken to avoid redical sign of square root).
Or, 3x^2 dx = 2udu or, x^2 dx =^23 udu
I u.udu^2
= 3 ( )
as 1 x u u+^3 =^2 =

( ) ( )

2 2 2 u 2^33323 3/2

= 3 u du .=3 3 9= 1 x+ = 9 1 x+ +c.

Example 136: 2

xdx.

(^) 3x 1+ Let 3x
(^2) +1 = u (^2) ; 6xdx = 2udu, xdx =^1
3 udu.
I^1 udu 1 du u 3x 1 c.1 1^2
=3 u 3 = =3 3= + +