138 Chapter 5Integration
EXAMPLES 5.8Improper integrals
(i) The function is not defined at x 1 = 1 0 and the definite integral between limits
x 1 = 1 0 and x 1 = 1 1 is defined as
Then
and, letting ε 1 → 1 0,
(ii)
The limit does not exist because ln 1 ε 1 → 1 −∞as ε 1 → 1 0.
(iii) In the general case of the integral of an inverse power, a 1 ≠ 11
Whena 1 < 11 the limit has value 12 (1 1 − 1 a), but whena 1 > 11 the limit is infinite
and the integral is not defined.
0 Exercise 35
Infinite integrals
It often happens in applications in the physical sciences that one or both of the
limits of integration are infinite. Integrals with infinite ranges of integration are
called infinite integrals. If the upper limit is infinite then the definite integral is
defined by
(5.19)
ZZ
aabfxdx fxdx
b
∞∞
() lim ()=
→
Z
01110
1
1
11
1
dx
x
a
x
aa=
−
=
→ −
−lim
εεaa
a
−
→
−lim
εε
0
1
1
1ZZ
011100
dx
x
dx
x
==x
=
→→ →
lim lim ln lim
εε εεε
00(ln)− ε
Z
012
dx
x
=
ZZ
εεεε
1112112 12222
dx
x
===−xdx x
−ZZ
0110
dx
x
dx
x
=
→
lim
εε1 x