EXAMPLE 16
Find the length of the arc of y = ln sec x from x = 0 to
SOLUTION:D. IMPROPER INTEGRALS
There are two classes of improper integrals:
(1) those in which at least one of the limits of integration is infinite (the interval is not bounded); and
(2) those of the type where f (x) has a point of discontinuity (becoming infinite) at x = c, a c
b (the function is not bounded).
Illustrations of improper integrals of class (1) are:
The following improper integrals are of class (2):Sometimes an improper integral belongs to both classes. Consider, for example,In each case, the interval is not bounded and the integrand fails to exist at some point on the interval
of integration.
Note, however, that each integral of the following set is proper:
The integrand, in every example above, is defined at each number on the interval of integration.