Graphics Reference
In-Depth Information
When a function
f
is integrable on the interval [
a
,
b
] with integral
I
,
and
I
L
as
b
, wewrite
f
L
. When the limit exists,
f
is
called an
improper integral
.
In qns 5.56
—
5.61 there are theorems which show that the existence
of an improper integral may be equivalent to the convergence of an
infinite series.
62 Givea definition for
f
analogous to that above. Illustrate your
definition with an example. When the limit exists,
f
is called an
improper integral
.
63 If
a
and
b
arepositivenumbrs, find thevalueof
f
where
f
(
x
)
1/
x
.
Determine lim
f
.
When a function
f
is integrable on the interval [
a
,
b
] with integral
I
,
and
I
L
as
a
c
, wewrite
f
L
, even when
f
is not integrable on
[
c
,
b
]. When this limit exists
f
is called an
improper integral
.
dx
64
Find
x
)
as an improper integral.
(1
dx
(1
x
)
dx
(1
x
)
.
If
1
a
0, show that 0
dx
(1
x
)
exists as an improper integral.
Deduce that
65 Givea definition for
f
when
f
is integrable on [
a
,
b
] but not on
[
a
,
c
], analogous to that above.
Illustrateyour definition with an example.
dx
(1
x
)
as an improper integral.
66
Find
dx
dx
If 0
a
1, show that 0
)
x
)
.
(1
x
(1
dx
(1
x
)
exists as an improper integral.
Deduce that
67 A function
f
is defined for
x
0by
(
1)
n
1
when
n
x
n
1;
n
0,1, 2, . . .
f
(
x
)
Do either (or both) of
