7.8 Improper Integrals

7.8 Improper Integrals - 7.8 Improper Integrals Definition...

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7.8 Improper Integrals Definition: The definite integral is called an improper integral if (a) At least one of the limits of integration is infinite, or (b) The integrand f ( x ) has one or more points of discontinuity on the interval [ a , b ]. b a dx x f ) (
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Infinite Limits of Integration 1. If f is continuous on the interval , then 1. If f is continuous on the interval , then 1. If f is continuous on the interval , then ) , [ a In each case, if the limit exists, then the improper integral is said to converge ; otherwise, the improper integral diverges . In the third case, the improper integral on the left diverges if either of the improper integrals on the right diverges. ) , ( -∞ ] , ( b -∞ = a b a b dx x f dx x f ) ( lim ) ( - -∞ = b b a a dx x f dx x f ) ( lim ) ( - - + = c c dx x f dx x f dx x f ) ( ) ( ) ( where c is any real number
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Discontinuous Integrand 1. If f is continuous on the interval and has an infinite discontinuity at b , then 1. If f is continuous on the interval and has an infinite discontinuity at a, then 1. If f is continuous on the interval
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