440 Chapter 7 • The Riemann Integral
(Of course, if la+oo f diverges, then so does la+oo g.)Proof. Exercise 8. •Theorem 7.8.14 (Comparison Test, Ilb) Suppose that \:/x ~ b, 0 <
f(x) ~ g(x). If 1~ 00 g converges, then so does 1~ 00 f, and 1~ 00 f~1~ 00 g.
(If 1~ 00 f diverges, then so does 1~ 00 g.)Proof. Exercise 9. •Example 7.8.15 j+oo f dx converges, since Vx :'.:". 1, f
1 x +5 x + 5
1 j+oo 1(^2) x , and 1 2 x dx was shown to converge in Example 7.8.12.^0
< x
x3
Theorem 7.8.16 (Absolute Convergence) Suppose 1: f {where a= oo or
b = -oo) is an improper integral of Type II. If 1: If I converges, then so does
1: f. [In this case, we say that 1: f converges absolutely.]
Proof. Exercise 10. •
j
Example 7.8.17 +oo sinx I sinx I
1
~ dx converges a bsolutely, since \:/x :'.:". 1, ~ ~
1 j+oo 1
(^2) x , and 1 2 x dx was shown to converge in Example 7.8.12.^0
Note: For an example showing that convergence does not imply absolute
convergence, see Exercise 5.
IMPROPER INTEGRALS OF MIXED TYPES
Improper integrals can be of mixed types, as the following example shows.
In such cases, we split the improper integral into two or more improper integrals
and employ the appropriate methods from each type as needed.
Example 7.8.18 Determine the convergence or divergence of {+oo ~ dx.
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