1549901369-Elements_of_Real_Analysis__Denlinger_

7.8 *Improper Riemann Integrals 435

l: f an improper integral of type I. If lim _t exists (as a real number)

c->b- a

then we say that the improper integral converges, and we write

t a f = c-+lim b-_t a f.

If this limit does not exist, then we say that the improper integral diverges.

In Definitions 7.8.1 and 7.8.2, :le> 0 such that either f is unbounded on

[a, a + c) or f is unbounded on (b -c, b], but not both (see Exercise 1). We can

extend the notion of improper integral to include cases in which both of these

are true, or f is unbounded in some neighborhood of an interior point of [a, b].

The following definition covers these possibilities.

Definition 7.8.3 Suppose that, for some a < c < b, one or both of l: f

and J: f are improper integrals in a sense previously defined. If the improper

integrals (both) converge, t hen we say that l: f converges, and write

Otherwise, we say that l: f diverges.

Example 7.8.4 Show that each of the following is an improper integral, and

determine their convergence or divergence.

(a) fl~ dx

l o x

fl 1
(b) lo Vx dx
1
Solution. (a) The function f(x) 2 is continuous and bounded on
x
any closed interval [c, 1], for 0 < c < l. Thus, f is integrable on [c, 1], but is
not integrable on [O, 1] since it is not bounded there (it is not even defined at

x = 0). Thus, f

1

~ dx is an improper integral.

lo x

1

1

Now, for 0 < c < 1,^1 [ -li c^1

c x^2 dx = - x^1 = -c - l.

Thus, lim 1

1

~ dx = lim (~ - 1) = +oo. Therefore, fl~ dx diverges.

c->O+ c X c-.o+ C lo X

1

(b) The function f ( x) = Vx is continuous and bounded on any closed

interval [c, 1], for 0 < c < l. Thus, f is integrable on [c, 1], but is not integrable
on [O, 1] since it is not bounded there (it is not even defined at x = 0). Thus,

fl 1 d... 1
lo Vx x is an improper mtegra.