Advanced Calculus fi..

452 Advanced Calculus, Fifth EditionRemark.It follows from Theorem 52 that for a convergent integral Xrn* f (x) dx,for every fixed k. However, it is not necessary that f (x) itself approach 0. This isillustrated by the integral (Problem 1 following Section 6.25):sin x2 dx.lWTHEOREM 53 Let f (x) be continuous for a 5 x c oo. 1f Jam If (x)l dx converges,thenconverges, andIn words: An absolutely convergent improper integral is convergent.Proof. Since If 1 is a continuous function of f, If (x)l is a continuous function ofa continuous function and must hence be continuous. NowHence if the Cauchy criterion holds for the integral of If 1, it must also hold forthe integral of f itself; thus the convergence of j" If 1 dx implies that of J f dx.Furthermore,This is true for every b; hence in the limit,THEOREM 54 (Comparison Test) Let f(x) and g(x) be continuous fora 5 x c oo. If 0 5 If (x)l 5 g(x) and Jam g(x) dx converges, thenis absolutely convergent, and