James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

528 Chapter 7 Techniques of Integration
Using this example as a guide, we define the integral of f (not necessarily a positive
function) over an infinite interval as the limit of integrals over finite intervals.
1 Definition of an Improper Integral of Type 1
(a) If y t a
f sxd dx exists for every number t > a, then
y`
a
f sxd dx − lim
t l ` yt f sxd dx
a
provided this limit exists (as a finite number).
(b) If y b f sxd dx exists for every number t < b, then
t
y b 2`
f sxd dx − lim
t l2` yb f sxd dx
t
provided this limit exists (as a finite number).
The improper integrals y`
f sxd dx and a yb f sxd dx are called convergent if the
2`
corresponding limit exists and divergent if the limit does not exist.
(c) If both y`
f sxd dx and a ya f sxd dx are convergent, then we define
2`
y`
2`
f sxd dx − y a 2`
f sxd dx 1 y`
f sxd dx
In part (c) any real number a can be used (see Exercise 76).
a
Any of the improper integrals in Definition 1 can be interpreted as an area provided that
f is a positive function. For instance, in case (a) if f sxd > 0 and the integral y`
f sxd dx
a
is convergent, then we define the area of the region S − hsx, yd | x > a, 0 < y < f sxdj
in Figure 3 to be
This is appropriate because y`
a
of f from a to t.
y
AsSd − y`
a
f sxd dx
f sxd dx is the limit as t l ` of the area under the graph
y=ƒ
S
FIGURE 3
0
a
x
ExamplE 1 Determine whether the integral y`
1
SOLUTION According to part (a) of Definition 1, we have
y`
1
1
1
dx − lim
x t l ` yt dx − lim
1 x ln | x t
|g 1
t l `
s1yxd dx is convergent or divergent.
− lim
t l `
sln t 2 ln 1d − lim
t l `
ln t − `
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