Improper Integral
Since the concept of Laplace transform involves an integral from zero to infinity, the
knowledge of improper integral is needed.
Definition
: An improper integral over an unbounded interval is defined as a limit of
integrals over finite intervals; thus
f(t)dt
a
!
"
=
lim
A
#!
f(t)dt
a
A
"
where A is a positive real number.
If the integral from a to A exists for each A > a, and if the limit as A
→
∞
exists, then the
improper integral is said to
converge
to that limiting value. Otherwise the integral is said
to
diverge
, or fail to exist,
Examples
:
1) Let f(t) = e
ct
, t
≥
0 and c is a real nonzero constant.
e
ct
dt
=
lim
A
!"
e
ct
dt
=
lim
A
!"
0
A
#
0
"
#
e
ct
c
0
A
=
lim
A
!"
1
c
e
cA
$
1
( )
if c < 0 , e
cA
→
0 as A
→
∞
, then the improper integral converges to 1/c,
if c > 0, e
cA
→
∞
as A
→
∞
, then the improper integral diverges,
if c = 0 , e
ct
= 1, then the improper integral diverges.
2) Let f(t) = 1/t, t
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 Fall '08
 STAFF
 Calculus, Topology, Continuous function, improper integral diverges

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