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Indeterminate Forms
                     CHAPTER 5







                                                     Fig. 5.1                                    133

                     5.3.2     INTEGRALS WITH INFINITE INTEGRANDS
                     Let f be a continuous function on the interval [a, b) which is unbounded as
                     x → b (see Fig. 5.2). The integral
                           −
                                                   b

                                                     f(x) dx
                                                  a



















                                                     Fig. 5.2
                     is then called an improper integral with infinite integrand at b. We often just say
                     “improper integral” because the source of the improperness will usually be clear
                     from context. The next definition tells us how such an integral is evaluated.
                        If
                                                   b

                                                     f(x) dx
                                                  a
                     is an improper integral with infinite integrand at b then the value of the integral is
                     defined to be
                                                     b−(
                                              lim       f(x) dx,
                                             (→0 +  a
                     provided that this limit exists. See Fig. 5.3.
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