1549901369-Elements_of_Real_Analysis__Denlinger_

442 Chapter 7 • The Riemann Integral

(a)

1

(^00) dx
1 x3
1
(^00 1)
(g) --dx
o 1 + x2
J
oo 1
(i) --dx
-oo 1 + x2
roo x
(m) 11 (x2 - 1 )3 dx
(d) roo dx
lo ,/ex
(f) fo
00
sin x dx
(h) l oo e-; dx
(j) {oo e -ft dx
lo Vx
(1)!
00
x dx
11 Jx^2 - 1
{ 00 1
(n) 11 x (lnx)2 dx

2. For each of the following, determine the values of r for which the integral
   exists or converges, and determine the values of the integral in those cases.
   (a) J 100 x -rdx (b) J 01 x -rdx (c) J 000 x-rdx


3. Use the comparison test to determine whether the following improper
   integrals converge:
   (a) roo dx
   11 Jx^3 + x

(b) {

00

xdx

11 Jx^3 + x

(d) {oo dx

11 xv'x+i

J

oo dx

(f) _ 00 x^2 + 4x + 6

4. Find a function f such that f 1
   00
   f converges, but f 1
   00
   VJ does not.


5. A classic example of an improper integral that converges, but not abso-
   . 1

00

lutely, is -sinx - dx. Venfy. t his,. as follows:

7r x

1

(^00) cosx
(a) Prove that - 2 - dx converges (absolutely).
7r x
(b) Use. mtegrat10n. by parts and (a) to prove that^1 oo~nx --dx converges.
7r x