Example 21 __
Example 22 __
.Thus, this improper integral diverges.
Example 23 __
. Since this limit does not exist (sin b takes on values
between −1 and 1 as b → ∞), it follows that the given integral diverges. Note,
however, that it does not become infinite; rather, it diverges by oscillation.
Improper integrals of class (2), where the function has an infinite
discontinuity, are handled as follows.
To investigate , where f becomes infinite at x = a, we define
. The given integral then converges or diverges according
to whether the limit does or does not exist. If f has its discontinuity at b, we
define ; again, the given integral converges or diverges as
the limit does or does not exist. When, finally, the integrand has a discontinuity
at an interior point c on the interval of integration (a < c < b), we let
.
Now the improper integral converges only if both of the limits exist. If either
limit does not exist, the improper integral diverges.
The evaluation of improper integrals of class (2) is illustrated in Examples 24–