dx is convergent if b xe x is convergent. b dx 0 x xe 0 Using integration by parts 0 l lim x x b xe + 0 dx b e 0 x b = 1 lim  b e b b 87 Hence x is convergent. xe dx 0 b 9.7 Theorem: if ( )dx f x is an absolutely convergent improper a integral it is also convergent, where b can be finite or infinite. By Cauchy’s inequality, a b b ( ) f x dx ( ) f x dx a b If ( ) f x dx is convergent (or absolutely convergent) then it has an upper a bound M ( )dx f x M b Hence ( )dx b f x a M and hence it is convergent. Then a
0 xe x dx is convergent if b b 0 xe x dx is convergent. Using integration by parts lim b l 0 xe x x b 0 + b e 0 x dx = lim b  1 b b 87 Hence 0 xe x dx is convergent. 9.7 Theorem: if b a f ( x )dx is an absolutely convergent improper integral it is also convergent, where b can be finite or infinite. By Cauchy’s inequality, b a f ( x ) dx b a f ( x ) dx If b a f ( x ) dx is convergent (or absolutely convergent) then it has an upper bound M Hence Then b a b a f ( x )dx f ( x )dx M M and hence it is convergent. e
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