Definition 8.9.4
If

is continuous on the interval

with an infinite limit on the left at

, then
If
is continuous on the interval
with an infinite left on the right at
, then
In both cases, if the limit exists and is finite, the improper integral is convergent. Otherwise, it is divergent.
Example 8.9.5
Let

.
We have:
Thus:
The given improper integral is divergent.
Example 8.9.6
Let

.
We have:
Thus:
Definition 8.9.7
Suppose that the function

is continuous at all the points of
![$ [a,b]$](img825.png)
, but not at the interior point

.
Moreover suppose that

has an infinite (at least one-sided) limit at

. Then:
where the two improper integrals are defined as in Def.
9.4.
Example 8.9.8
Let

.
By definition, we have:
We compute each of the terms:
Thus:
By the same way we show that:
Therefore the given improper integral is divergent.
Noah Dana-Picard
2007-12-28