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IMPROPER INTEGRALS
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IMPROPER INTEGRALS OF THE SECOND KIND If f x becomes unbounded only at the end point x a of the interval a @ x @ b, then we de ne b b f x dx lim f x dx 4
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and de ne it to be an improper integral of the second kind. If the limit on the right of (4) exists, we call the integral on the left convergent; otherwise, it is divergent. Similarly if f x becomes unbounded only at the end point x b of the interval a @ x @ b, then we extend the category of improper integrals of the second kind. b  b f x dx lim f x dx 5 !0 a a 1 sin x dx Note: Be alert to the word unbounded. This is distinct from unde ned. For example, 1 0 x sin x sin x lim dx is a proper integral, since lim 1 and hence is bounded as x ! 0 even though the !0  x x!0 x function is unde ned at x 0. In such case the integral on the left of (5) is called convergent or divergent according as the limit on the right exists or does not exist. Finally, the category of improper integrals of the second kind also includes the case where f x becomes unbounded only at an interior point x x0 of the interval a @ x @ b, then we de ne b x0 1 b f x dx lim f x dx lim f x dx 6
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The integral on the left of (6) converges or diverges according as the limits on the right exist or do not exist. Extensions of these de nitions can be made in case f x becomes unbounded at two or more points of the interval a @ x @ b.
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CAUCHY PRINCIPAL VALUE It may happen that the limits on the right of (6) do not exist when 1 and 2 approach zero independently. In such case it is possible that by choosing 1 2  in (6), i.e., writing & x0  ' b b f x dx lim f x dx f x dx 7
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the limit does exist. If the limit on the right of (7) does exist, we call this limiting value the Cauchy principal value of the integral on the left. See Problem 12.14.
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EXAMPLE. The natural logarithm (i.e., base e) may be de ned as follows: x dt ; 0<x<1 ln x 1 t 1 1 Since f x x is unbounded as x ! 0, this is an improper integral of the second kind (see Fig. 12-2). Also, dt is an integral of the third kind, since the interval to the right is unbounded. 1 0 t dt lim ln 1 ln  ! 1 as  ! 0; therefore, this improper integral of the second kind is Now lim !0  t 1 !0 x dt dt divergent. Also, lim lim ln x ln i ! 1; this integral (which is of the rst kind) also diverges. x!1 1 t x!1 t 1
CHAP. 12]
IMPROPER INTEGRALS
Fig. 12-2
SPECIAL IMPROPER INTEGRALS OF THE SECOND KIND b dx 1. p converges if p < 1 and diverges if p A 1. a x a b dx 2. p converges if p < 1 and diverges if p A 1. a b x These can be called p integrals of the second kind. Note that when p @ 0 the integrals are proper.
CONVERGENCE TESTS FOR IMPROPER INTEGRALS OF THE SECOND KIND The following tests are given for the case where f x is unbounded only at x a in the interval a @ x @ b. Similar tests are available if f x is unbounded at x b or at x x0 where a < x0 < b. 1. Comparison test for integrals with non-negative integrands. (a) Convergence. Let g x A 0 for a < x @ b, and suppose that b 0 @ f x @ g x for a < x @ b, f x dx also converges.