7.8 Improper Integrals
In this section we will be looking at definite integrals where (1) the interval is infinite and (2)
the function is unbounded (an infinite discontinuity). These are called
improper integrals.
Type 1: Infinite Integrals - Definition on page 524.
We can use an integral to find area under a curve over a closes interval, but what about an
infinite interval? It may be possible to evaluate the area and determine that the integral is
convergent.
However, it may not be possible to find an area. In this case the integral is
divergent.
We will examine
Example 1
from the text. Why does one function converge and the other does
not. These functions seem to be very similar. Limits are used to calculate improper integrals.
So, why do we get different results for such similar functions?
1
( )
f x
x
=
just doesn’t converge “fast enough”.
When evaluating the limits, all past techniques (i.e. L’Hospitals Rule, etc) for limits and

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