Advanced Calculus fi..

*I. 421 8 Advanced Calculus, Fifth EditionIf the Riemann integral of f fails to exist, one may still be able to obtain a valueby treating the integral as an improper integral. For example,3does not exist as a Riemann integral for p > 0, since the integrand is unbounded,and the integral is improper because of the discontinuity at x = 0. One seeks a valueby a limit processFor p = 1 the integral from b to 1 has the value -log b, so that there is no limit asb + 0. Otherwise, the integral has value@1 - Pand this has limit 1 /(I- p) for 0 < p < 1 but has no limit for p > 1. Hence one writesand assigns no value for p 2 1. Here the improper integral is said to converge for0 < p < 1 and to diverge for p > 1.If there are several discontinuities (a finite number) on the interval, one breaksthe integral up into integrals with just one discontinuity; if each of these converges,then the given integral is termed convergent and assigned as value the sum of thevalues of the partial integrals; otherwise, it is termed divergent. For example, for anintegral from 0 to 5 with discontinuities at 1 and 4, one considers the integrals from0 to 1, 1 to 3, 3 to 4, and 4 to 5.Integrals with infinite limits are also improper, and one evaluates them by a limitprocess. For example,Lmw 0 03xe-" dx = J__ xe-x2 dx + l xc"1 K .dx = lim arc tan b = -,b-t m 2dxe-~2 0 :t '1= lim -b-,-w -2+ lim -b+m -2Improper integrals are discussed further in sections 6.22 to 6.24.Numerical evaluation of dejnite integrals. The formula (4.11) is often of no help,since the primitive F is hard to find (or may not exist in usable form). Hence inpractice, many definite integrals are evaluated numerically with the aid of computers.Sophisticated procedures to this end have been developed. We mention here only