## Second type.

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 , 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