The Chemistry Maths Book, Second Edition

5.3 The definite integral 137

EXAMPLE 5.7The function

is continuous at x 1 = 10 , but not smooth, the slope of its

graph changing discontinuously from + 1 to − 1 on passing

throughx 1 = 10 from left to right (Figure 5.8). The function

can be integrated if the range of integration is split at the

point of gradient discontinuity:

0 Exercises 32–34

Improper integrals

A definite integral is called improperwhen the integrand has an infinite discontinuity

at a point within the range of integration. If the discontinuity is at the pointx 1 = 1 c,

wherea 1 ≤ 1 c 1 ≤ 1 b, then the integral is defined as the limit, forε 1 > 10 ,

(5.17)

As shown in Figure 5.9, the point cis excluded because the integrand is not defined

there. When the limit in (5.17) is finite and unique then the value of the integral is the

‘area under the curve’.

If the discontinuity lies at one end of the range of integration, atx 1 = 1 asay, the integral

is defined as

(5.18)

ZZ

a

b

a

b

fxdx() lim= fxdx()

→

+

ε

ε

0

ZZZ

a

b

a

c

c

b

fxdx() lim=+fxdx() fxdx()









→

−

+

ε

ε

ε

0













=−

( )

+−

( )

=− −

−−−−

112 eeee

abab

ZZZ

−

−

−

−

−

=+=













• 
  

a

b

x

a

x

b

x

a

x

edx edx edx e

||

0

0

0

0

bb

x

−e













−

fx e

ex

ex

x

x

x

()

||

==

≥

≤











−

−

if

if

0

0

.....

........

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0

1

−a

b

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...

.

Figure 5.8

ab−ccc +

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Figure 5.9