We have considered only integrals of the form
where
are real numbers and
is a function which is
defined and continuous on the interval
.
Sometimes, it makes sense to consider integrals over infinite intervals and for functions that are discontinuous or not necessarily defined at every point in the interval.
What sense can we make of
?
The function is positive for every value of
.
Thus,
ought to be the area of the
region bounded by the graph of
, the
-axis, and the
-axis.
This region is eventually covered by the regions bounded by
, the
-axis,
-axis, and the line
for
a sufficiently large real number.
In this case,
.
If is a real number and
is a function which is continuous
on the interval
, then
we define
.
Nota Bene: This limit might not exist!
Compute
.
This limit does not exist! For each value of , there are
and
bigger than
with
and
(take
to be
an even multiple of
and
an odd multiple of
).
Compute
.
Compute
.
Via the change of variables (with
), we
see that
Analogously to integrals of the form
, we
define
.
If this limit does not exist, then we say that the integral is undefined.
We define
.
Nota Bene: There are two separate limits involved in the
definition of
. Namely,
If the limits defining
exist, then
However, the limit on the righthand side of this equation may exist without
being defined.
Compute
.
That is, the limit does not exist. Therefore,
is
undefined.
However,
.
If is continuous for
, then we define
. When
is continuous at
as well, then
this definition agrees with the old definition.
Compute
.