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Unformatted text preview: Section 8.8: Improper Integrals An improper integral is a definite integral in which the integrand f x ( 29 has a discontinuity on the interval a , b [ ] or one in which the interval involve . Improper Integrals of Type I: (i) If f x ( 29 is continuous on a , [ 29 (in other words, the interval extends to ), then f x ( 29 dx a = lim t f x ( 29 dx a t (ii) If f x ( 29 is continuous on - , b ( ] (in other words, the interval extends from - ), then f x ( 29 dx- b = lim t - f x ( 29 dx t b (iii) If f x ( 29 is continuous on - , ( 29 (in other words, the interval extends from - and to ), then f x ( 29 dx- = f x ( 29 dx- c + f x ( 29 dx c = lim t - f x ( 29 dx t c + lim t f x ( 29 dx c t where c is any real number. In each case, if the limit is finite then the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges....
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