1549901369-Elements_of_Real_Analysis__Denlinger_

7.8 *Improper Riemann Integrals 437

Proof. Suppose l: f is an improper integral of Type I. We consider the
case in which f is integrable over every subinterval [c, b], where a < c < b. The
other cases have similar proofs. Now, \:/x E (a , b],

-lf(x)I :S f(x) :S lf(x)I, so

0 :S f(x) + lf(x)I :S 2lf(x)I. (45)

But l: Iii converges, by hypothesis, so l: 2lfl also converges. Hence, by

(45) and the comparison test, l: (f(x) + lf(x)I) converges.

Now, \:/a < c < b,

l: f = f:[U + lfl) - lfll = J:u + lfl) - J: Iii-

Then lim J: f exists, since both lim l:U + lfl) and lim l: Iii exist.
c--+a+ c--+a+ c--+a+
Therefore, l: f converges. •

EXERCISE SET 7.8-A

l. Suppose the hypotheses of Definition 7.8.1 are met: a < b and f is inte-

grable on every closed subinterval of the form [c, b], where a < c < b. But

suppose that f is not integrable on [a, b]. Prove that 3 c > 0 3 f is not

bounded on [a, a+ c). State a similar result about Definition 7.8.2.

2. Prove that if f is integrable on [a, b], where a < b, then lim l: f = l: f
   c-.a+
   and c->lim b-;:c a f = t a f.


3. In each of the following, determine whether l: f is an improper integral
   and, if so, determine its convergence or divergence. When possible, find
   the values of those improper integrals that converge.

1

1 1

(a) - dx

0 x

fl 1

(b) Jo v'xdx 1

1 1

(c) - dx

o ex

1

2 1

(d) -

1

-dx

1 x nx

(e) fo

1

lnxdx (f)^13 x dx

l~