8.8 - 8.8 1 8.8 2 8.8 Improper Integrals Type I - Infinite...

8.8 1
8.8 Improper Integrals
Type I - Infinite Limits of Integration
Consider the following example.
Example 1. Infinite Areas?
Suppose that b > 1 . Evaluate the following
definite 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.