Calculus: Analytic Geometry and Calculus, with Vectors

12.2 Ratio test and integral test 609

by choosing a number r for which 1 < r < p and showing that there is a con-
stant 14 for which

0<(xp =(log x)Q 1^511
xp-' x' - Xr
when x > 2. Prove that
f°° log x
3 x

21 Using results of the preceding problem when and if they are helpful, prove

that the first of the integrals

1

XP fr

o xP r m x' dx

(1 x)4

dx,

Ji (1 + x)adx, Jo (1 + x)°

exists when p > -1 and fails to exist when p 5 -1. Prove that the second

integral exists when q - p > 1 and fails to exist when q - p < 1. For what

pairs of values of p and q does the third integral exist?

22 The first of the two integrals

(1) f¢mf(x) dx, fa f(x)I dx

is sometimes said to converge absolutely if the second one exists. Prove the fol-

lowing theorem.

(2) Theorem If f is Riemann integrable over each finite interval a < x <= h

for which h > a and if fa I f(x)j dx < w, then fa f(x) dx exists.

Solution: This theorem and its proof are very similar to Theorem 12.17 and

its proof. Let

p(x) = _[I f(x)I + f(x)), q(x) = [If(x)I - f(x)],

so that 0 < p(x) S Jf(x)I and 0 < q(x) <_ f(x)I. It then follows from the

theorem of Problem 20 that the limits in

lim foh p(x) dx = L1, lim Jo q(x) dx = L2

h- - h-,.

lim foh f(x) dx = 1im [ foh p(x) dx-Joh q(x) dx,

h- - h- -

23 Does existence of f 0I f f (x) I dx imply existence of f o- f (x) dx? -4,s.:

No. For example, f(x) might bees when x is rational and -e- when x is

irrational. In this case If(x)I = e-2 and

fo'

If(x)I dx = 1 but f is everywhere

discontinuous and there is no interval over which f is Riemann integrable.

24 Prove that if f and g are both Riemann integrable over each finite interval

a 5 x <- h for which h > a, and if

If(x)I < Ig(x)I (x > a)