Paper 4: Fundamentals of Business Mathematics & Statistic

3.64 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Agian let x = φ (z), Then dxdz=φ ′ (z)., (change of variable)

Now, dz dx dzdI dI dx=. = f (x). φ ′ (z) = f [ φ (z)]. φ ′ (z)

∴ by def. I = f [φ (z)] φ ′ (z) dz if x = φ (z).

The idea will be clear from the following example :

Integrate. (2 + 3 x)n dx. Let 2 + 3 x = z ∴3 dx = dz

or, dx 1dz 3= (i.e. here φ ′ (z) = dx 1dz 3= )

Now, I = zn. 31 dz, as f [ φ (z) = zn.

1

3 z

n dz =

1 zn^11

3 n 1 3(n 1).

+

+ = + (2 + 3x) n +1 + c. (putting z =2 + 3x)

Note. It may be noted that there is no fixed rule for substitution in solving these types of problems.

Important Rules

1. ( ) e

f (x)dx log f(x).

f x

′ =

2. ( )
   f(x) .f (x)dxp^1 f x ,(x 1).p 1
   p 1

′ =^ + ≠ −

(^) +

3. e f x f (x) dx e f(x).x^ ( )+ =′ x

Example 127: (^) 2 xd+x =log(2 x).+ Let 2+x = z, dx = dz.
I dz log z Log(2 x)as dx log x.
= (^) z = = + (^) x =
Example 128: (2 x ) .2xdx+ 2 3
Let 2 + x^2 = z
2x dx = dz
I z dz z^314
= = 4
(^1) (2 x )2 4
= 4 +
Example 129: e (x 1)dxx +
Let, ex. x = z
(ex + ex. x) dx = dz
or, ex (x+1) dx = dz
I dz z e x= = = x⋅