关于数学分析的证明题I(积分)

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$f命题：$设$fleft( x ight),varphi left( x ight)$在任何有限区间$[a,b]$上可积，且$varphi left( x ight) > 0left( {x > a} ight),int_a^{ + infty } {varphi left( t ight)dt} $发散，$fleft( x ight) = oleft( {varphi left( x ight)} ight)left( {x o  + infty } ight)$，证明：$int_a^{ + infty } {fleft( t ight)dt}  = oleft( {int_a^{ + infty } {varphi left( t ight)dt} } ight)$

$f命题：$