Barrons AP Calculus - David Bock

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 to be 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 to be 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 24–31.
BC ONLY

EXAMPLE 24

Find

SOLUTION:

In Figure N7–24 we interpret this integral as the first-quadrant area under and to the left of x =

1.

FIGURE N7–24

EXAMPLE 25

Does converge or diverge?

SOLUTION:

Therefore, this integral diverges.