Page 316 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 12] IMPROPER INTEGRALS 307

(c)If f is integrable on 1 < x < 1, then
ð ð a ð
1 1
f ðxÞ dx
f ðxÞ dx ¼ f ðxÞ dx þ
a
1 1
ð a ð x
¼ lim f ðtÞ dt þ lim f ðtÞ dt:
x a
x! 1 x!1
In part (c)itis important to observe that
ð a ð x
lim f ðtÞ dt þ lim f ðtÞ dt:
x! 1 x x!1 a
and
ð a ð x
lim f ðtÞ dt þ f ðtÞ dt
x a
x!1
are not necessarily equal. 2
x
This can be illustrated with f ðxÞ¼ xe . The first expression is not defined since neither of the
improper integrals (i.e., limits) is defined while the second form yields the value 0.
1 2
EXAMPLE. The function FðxÞ¼ p ffiffiffiffiffiffi e ðx =2Þ is called the normal density function and has numerous applications
2
in probability and statistics. In particular (see the bell-shaped curve in Fig. 12-1)
ð 2
1 x
1
: dx ¼ 1
p ffiffiffiffiffiffi e
2 2
1
(See Problem 12.31 for the trick of making this evaluation.)
Perhaps at some point in your academic career you were
‘‘graded on the curve.’’ The infinite region under the curve with
the limiting area of 1 corresponds to the assurance of getting a
grade. C’s are assigned to those whose grades fall in a desig-
nated central section, and so on. (Of course, this grading
procedure is not valid for a small number of students, but as
the number increases it takes on statistical meaning.)
In this chapter we formulate tests for convergence or diver-
gence of improper integrals. It will be found that such tests and
proofs of theorems bear close analogy to convergence and Fig. 12-1
divergence tests and corresponding theorems for infinite series
(See Chapter 11).

CONVERGENCE OR DIVERGENCE OF IMPROPER INTEGRALS OF THE FIRST KIND
Let f ðxÞ be bounded and integrable in every finite interval a @ x @ b. Then we define
ð ð b
1
f ðxÞ dx ¼ lim f ðxÞ dx ð1Þ
a b!1 a
where b is a variable on the positive real numbers.
The integral on the left is called convergent or divergent according as the limit on the right does or
ð
1 1
X
does not exist. Note that f ðxÞ dx bears close analogy to the infinite series u n , where u n ¼ f ðnÞ,
ð b a n¼1
while f ðxÞ dx corresponds to the partial sums of such infinite series. We often write M in place of
a
b in (1).