1549901369-Elements_of_Real_Analysis__Denlinger_

434 Chapter 7 • The Riemann Integral

Theorem 7.7.35 (Characterization of the Cosine Function) The only

function F : JR ____, JR such that

(a) 'Vx E JR, F"(x) = -F(x),

(b) F(O) = 1, and

(c) F'(O) = 0

is the function F ( x) = cos x.

Proof. Exercise. •

7 .8 *Improper Riemann Integrals

In our definition of the Riemann integral of a function over an interval, we

require (1) that the function be bounded on the interval, and (2) that the in-

terval of integration be bounded. Indeed, the theorems we have given, and their

proofs, depend on these boundedness assumptions for their validity. There are

times, however, when we wish to extend the notion of integration to situa-

tions in which one or both of these boundedness assumptions are not met. For

example, since arcsin 1 = ~, we would expect that, in some sense,

fl dt

lo Vf=t2^2

(See Definition 7.7.18.)

even though the function 1/Vf=t2 is not Riemann integrable on [O, l]. As we

shall see below, this is an improper integral of "type I. "

IMPROPER INTEGRALS OF TYPE I

Definition 7.8.1 Suppose a < band f is integrable on every closed subinterval

of the form [c, b], where a < c < b, but f is not integrable on [a, b]. Then we call

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

C---ta+

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

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

Definition 7.8.2 Suppose a < b and f is integrable on every closed subinterval

of the form [a, c], where a < c < b, but f is not integrable on [a, b]. Then we call