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| PAGE: 551 | SET: Exercises | PROBLEM: 1 |
Definition of an improper integral :
(i)
If 
is continuous on 
and is discontinuous at 
, then 
if this limit exists (as a finite number).
(ii)
If 
is continuous on 
and is discontinuous at 
, then 
if this limit exists (as a finite number).
(a)
The integral is 
.
The function is 
.
The given integral is improper because has the vertical asymptote at 
.
Thus, the infinite discontinuity occurs at the left end point of 
.
(b)
The integral is 
.
.
The above integral is improper because upper integration limit is infinite.
This integral should be defined as a limit of proper integrals with finite integration range.
(c)
The integral is 
.
, where 
is arbitary.
The above integral is improper because bith upper and lower integration limits are infinite.
This integral should be defined as a limit of proper integrals with finite integration range.
(d)
The integral is 
.
The above integral is improper because the function 
is not continuous at point 
.
Therefore, convergence should be analyzed in the sense of a limit of proper integrals.
i.e, 
.
(a) Infinite discontinuity.
(b) Infinite interval.
(c) Infinite interval.
(d) Infinite discontinuity.
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