Question

求下列定积分的计算过程

书上考题答案是θ

Answer 1

∫x/θe^(-x/θ)dx  [0  +∞]

=-∫xe^(-x/θ)d(-x/θ)

=-∫xde^(-x/θ)

=-xe^(-x/θ)+∫e^(-x/θ)dx

=-xe^(-x/θ)-θe^(-x/θ) [0  +∞]

 

lim(x→∞)xe^(-x/θ)

=lim(x→∞)x/e^(x/θ)

=lim(x→∞)θ/e^(x/θ)

=0

 

原式=θ 

Answer 2

分部积分：θ应该大于0
原式=(-θ)∫(0,+∞)(x/θ)de^(-x/θ)
=(-θ)[(x/θ)e^(-x/θ)|(0,+∞)-∫(0,+∞)e^(-x/θ)d(x/θ)]
=(θ)[∫(0,+∞)e^(-x/θ)d(x/θ)]
=(-θ)[e^(-x/θ)| (0,+∞)
=θ