5.3 The definite integral 139
For example,
using the property of the exponential thate
−b1 → 10 asb 1 → 1 ∞. In practice, infinite
integrals are treated in exactly the same way as ordinary integrals, but some care must
be taken when assigning the infinite value to the variable.
EXAMPLE 5.9Find the value of the infinite integral.
The integrand can be written in terms of partial fractions (see Section 2.7) as
Then
because(b 1 − 1 2) 2 (b 1 − 1 1) 1 → 11 asb 1 → 1 ∞andln 111 = 10
0 Exercises 36–39
When the limit in the definition (5.19) is finite and unique, the integral is said to be
convergent. An integral is divergentwhen the limit is indeterminate. For example,
is divergent becauseln 1 b 1 → 1 ∞asb 1 → 1 ∞.
A more subtle example is
Z
00∞∞∞
cosxdx lim sinx lim sinb
bb
b=
=
→→
Z
11∞∞
dx
x
xb
bb
b=
=
→→
lim ln lim ln
∞
=
−
−
=
−
→→
lim ln lim ln
bb
bx
x
b
∞∞
42
1
2
bb−
−=
1
2
3
3
2
ln ln
I
xx
dx x
bb
bb=
−
−
−
=−
→→
lim lim ln(
∞∞
Z
441
2
1
1
2))ln( )−−
x 1
1
12
1
2
1
()( )xx x x−− 1
=
−
−
−
I
dx
xx
=
−−
Z
412
∞()( )
=−+
=
→
−lim
b
e
b∞
11
ZZ
000∞∞∞
edr edr e
rbrbrbb
−−−
==−→→
lim lim