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Background

Our basic theorem for is that the integral exists if is continuous on the closed interval . We have actually gone beyond this theorem a few times, and integrated functions that were bounded and had a finite number of jump discontinuities on . However, we don't have any theory to help us deal with integrals involving one or more of the following.

1. Functions , for example rational functions, that have vertical asymptotes in (or are not bounded on ).
2. Integrals where the interval is unbounded, for example intervals like , , or .

We have already seen at least one example of the problems you can run into if the function is unbounded. Recall the clearly absurd result

that is obtained by blindly applying the FTOC. The second type of problem, where the interval of integration is unbounded, occurs often in applications of calculus, such as the Laplace and Fourier transforms used to solve differential equations. It also occurs in testing certain kinds of infinite series for convergence or divergence, as we will learn later.