8.8
1
8.8
Improper Integrals
Type I - Infinite Limits of Integration
Consider the following example.
Example 1.Infinite Areas?Suppose thatb >1. Evaluate the followingdefinite integral.
(1)
integraldisplay
b
1
1
x
2
dx
y
= 1
/x
2
1
b
1
In other words, find the area under curve. It
follows that
integraldisplay
b
1
1
x
2
dx
=
-
1
x
b
1
= 1
-
1
b
(2)
8.8
2
Now, what happens in (2) as
b
→ ∞
? In this case
we have the following.
integraldisplay
∞
1
1
x
2
dx
= lim
b
→∞
integraldisplay
b
1
1
x
2
dx
= lim
b
→∞
1
-
1
b
= 1
Motivated by the previous example, we adopt the
following definition.
8.8
3
Definition.
Improper Integrals - Type I
Definite integrals with infinite limits of integration
are called
improper integrals of type I
. They
are defined as follows.
1. If
f
(
x
)
is continuous on
[
a,
∞
)
, then
integraldisplay
∞
a
f
(
x
)
dx
= lim
b
→∞
integraldisplay
b
a
f
(
x
)
dx
(3)
2. If
f
(
x
)
is continuous on
(
-∞
, b
]
, then
integraldisplay
b
-∞
f
(
x
)
dx
=
lim
a
→-∞
integraldisplay
b
a
f
(
x
)
dx
(4)
3. If
f
(
x
)
is continuous on
(
-∞
,
∞
)
, then
integraldisplay
∞
-∞
f
(
x
)
dx
=
integraldisplay
c
-∞
f
(
x
)
dx
+
integraldisplay
∞
c
f
(
x
)
dx
(5)
for any real number
c
. Notice that this last
case is handled by combining the first two.
8.8
4
In all three cases, we say the improper integral
converges
whenever the right-hand side is finite.
Otherwise, the improper integral
diverges
.
Remark.
So the improper integral in the first
example
converged
and its value was
1
. Also,
since
f
(
x
) = 1
/x
2
>
0
we can say that the “area
under the curve” is finite...in fact, the area is
1
.
8.8
5
The following result is very useful.
Proposition 1.
Suppose that
f
is continuous
on
[
a,
∞
)
and
integraltext
∞
a
f
(
x
)
dx
converges. Then
integraltext
∞
a
cf
(
x
)
dx
also converges for any
c
∈
R
.
Proof.
Let
b > a
. Then by the basic properties of
the definite integral
integraldisplay
b
a
cf
(
x
)
dx
=
c
integraldisplay
b
a
f
(
x
)
dx
It follows that
integraldisplay
∞
a
cf
(
x
)
dx
= lim
b
→∞
integraldisplay
b
a
cf
(
x
)
dx
=
c
lim
b
→∞
integraldisplay
b
a
f
(
x
)
dx
=
c
integraldisplay
∞
a
f
(
x
)
dx <
∞
8.8
6
Example 2.
Evaluate the improper integral
integraldisplay
∞
2
2
x
2
-
1
dx
Notice that the integrand is continuous on
[2
,
∞
)
so by (3) we have
integraldisplay
∞
2
2
x
2
-
1
dx
= lim
b
→∞
integraldisplay
b
2
2
x
2
-
1
dx
Now we can use the usual techniques (partial
fractions in this case) to evaluate the integral.
