$f引理：$设$int_a^{ + infty } {fleft( x ight)dx} $收敛，且${fleft( x ight)}$在$left[ {a,{ m{ + }}infty } ight)$单调，则$lim limits_{x o + infty } xfleft( x ight) = 0$，进而$lim limits_{x o + infty }fleft( x ight) = 0$
$f命题：$设$int_a^{ + infty } {fleft( x
ight)dx} $收敛，且可微函数${fleft( x
ight)}$在$left[ {a,{
m{ + }}infty }
ight)$单调递减，则$int_a^{ + infty } {xf'left( x
ight)dx} $收敛
$f证明$ 对任意的$x in left[ {a, + infty }
ight)$，由$f分部积分法$知
[int_a^x {tf'left( t
ight)dt} = tfleft( t
ight)left| {egin{array}{*{20}{c}}x\a
end{array}}
ight. - int_a^x {fleft( t
ight)dt} ]
而由$int_a^{ + infty } {fleft( t
ight)dt} $收敛知$lim limits_{x o + infty } int_a^x {fleft( t
ight)dt} $存在，又由引理知$lim limits_{x o + infty }xfleft( x
ight) = 0$，所以有$lim limits_{x o + infty }int_a^x {tf'left( t
ight)dt}$存在，从而由反常积分收敛的定义即证