58
Section
2:
Working
With
[F\n tails, like
that
of
f(x),
can be
truncated without greatly
degrading
the
accuracy
or
speed
of
integration.
But
g(x)
has too
wide
a
tail
to
ignore when calculating
t
s(x)dx
L
if
£is
large.
For
such functions,
a
substitution like
x = a + b tan u
works well,
where
a
lies within
the
graph's main "body"
and b is
roughly
its
width.
Doing
this
for
f(x)
from
above with
a = 0 and b =
I
gives
/
t
rtan~lt
f(x)
dx
=
I
e'ian\l
+
tan2u)du,
J
J
0
which
is
calculated readily even with
t as
large
as
1010.
Using
the
same substitution with
g(
x),
values near a = 0 and b =
10~5
provide
good
results.
This
example involves subdividing
the
interval
of
integration.
Although
a
function
may
have features
that
look
extreme over
the
entire interval
of
integration, over portions
of
that
interval
the
function
may
look
more well-behaved. Subdividing
the
interval
of
integration works best when combined with appropriate substitu-
tions. Consider
the
integral
I
dx/(l+x&4)
=
dx/(l+xM)
+
J
0
J
0
J
1
rl rl
=)
dx/(l
+
xM)+J
u6'2du/(u
Cl
=
/
J
0
rl
1
(x6
J
o
(
These steps
use the
substitutions
x =
I/
u and x =
vl/8
and
some
algebraic manipulation. Although
the
original integral
is
improper,
the
last
integral
is
easily handled
by
[7T|.
In
fact,
by
separating
the
constant term
from
the
integral,
you
obtain (using
|
SCI
1
8)
an
answer with
13
significant digits:
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