# prop_imp_ints.pdf - Proper and Improper Integrals Definite...

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Proper and Improper IntegralsDefinite integrals are defined as limits of Riemann sums (Stewart§5.2).Sometimes the limit process breaks down and integral expressions don’t ac-tually mean anything. Fortunately there is a theorem that shows this isnota problem for a big class of integrals.Definition: an integral expressionbaf(t)dtisproperif the limits and func-tion arebounded. More precisely,the limits of integration are finite (i.e. not±∞);the function is defined and bounded on the interval [a, b]; andthe function has only finitely many discontinuities in the interval.TheoremProper integrals are always well-defined (i.e. the limit converges).An integral is
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