Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.85

3.5.6 METHOD OF SUBSTITUTION

Rule of evaluate ( )
b

(^) af xdx by the substitution x = f (u) :

1. In the integral put x = φ (u) and dx = φ′ (u) and dx = φ′ (u) du.


2. From the relation x = f (u),
   For x = a, find the corresponding value of u say α.
   For x = b, find the corresponding value of u, say β.


3. Evaluate the new integrand with the new limits the value thus obtained will be the required value of
   the original integrand.

Note : In a definite integral substitution is reflected in three places :
(i) in the integrand, (ii)in the differential, and (iii) in the limits.
This idea will be clear from the following examples.

Example 175:Evaluate :

1

0 2

xdx

(^) 1 x+
Let 1 + x^2 = u^2 when x = 1, u^2 = 1 + 1 = 2 or, u = 2
or, 2xdx = 2udu when x = 0, u^2 = 1 + 0 = 1 or, u = 1
or, xdx = udu
(^2222)
1 2 1 1 1
I udu udu du u 2 1
∴ = (^) u = (^) u = = = −
Example 176:
1 7
0 8
x dx.
(^) 1 x+ Let 1 + x
(^8) = u, 8x (^7) dx = du
When x = 1, u = 1+1 = 2 ; x = 0, u = 1 + 0 = 1
( )
(^22)
1 1
I^1 du 1 log u^1 log 2 log1 log2.^1
∴ =8 u 8 = = 8 − = 8
Example 177:
b
a
logx .dx
(^) x Let log x = u,
dx du
x = , for x = b, u = log b
x = a, u = log a
( ) ( )
log b^2 logb 2 2
loga log a
I udu u^1 log b log a
2 2

∴ = =^ =^ −^

(^)
=^12 (logb loga log b loga+ ) ( − ) = 21 log ab log .( ) (^) ab (^)