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Definition A definite integral 
is called improper integral if the interval [a,b] of integration is infinite, or if 
is not defined or not bounded at one or more points in [a,b].
Example 
, 
, 
, 
are improper integral.
Definition (a) 
is defined as 
(b) 
is defined as 
(c) 
is defined as 
for any real number 
.
(Or 
)
(d) If is continuous except at a finite number of points, say 
where
, then is defined to be
for any 
such that 
.
Example Evaluate the integral
.
Solution By definition of an improper integral we find
.
Definition The improper integral is said to be convergent or divergent according to the improper integral exists or not.
Example Find out at which values of 
the integral
converges and at which diverges.
Example Find out at which values of the integral
converges and at which diverges.
Example For any non-negative integer 
, define
.
(a) Show if is positive integers, then 
.
Hence, if 
is convergent, 
is also convergent.
(b) Find .
Theorem Let and 
be two real-valued function continuous for 
. If 
then the fact that 
diverges implies 
diverges and the fact that converges implies that converges.
Example Investigate the convergence of the integral
.
Theorem Let 
is converges. Then 
is also converges.
In this case, the integral is called absolutely convergent.
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