题目

2道,微积分,
⑴ 2∫（下限是0,上限是正无穷大）*2x*e的（-2x）次方dx+3∫（下限是0,上限是正无穷大）*3x*e的（-3x）次方dx
⑵ ∫（下限是0,上限是正无穷大）*（∫（下限是0,上限是正无穷大）*xy*e的（-x-y）次方*dy）dx

提问时间：2020-12-26

答案

(1)2∫（下限是0,上限是正无穷大）*2x*e的（-2x）次方dx+3∫（下限是0,上限是正无穷大）*3x*e的（-3x）次方dx
= -2∫（下限是0,上限是正无穷大）*xd(e^(-2x))-3∫（下限是0,上限是正无穷大）*xd(e^(-3x))
= {-2xe^(-2x)|(0,无穷）+2∫（下限是0,上限是正无穷大）e^(-2x)dx}+{-3xe^(-3x)|(0,无穷）+3∫（下限是0,上限是正无穷大）e^(-3x)dx}
= {0+[-e^(-2x)]|(0,无穷)}+{0+[-e^(-3x)]|(0,无穷)}
={1}+{1}
=2
(2)此类属于变量可分离的积分；
∫（下限是0,上限是正无穷大）*（∫（下限是0,上限是正无穷大）*xy*e的（-x-y）次方*dy）dx
={∫（下限是0,上限是正无穷大）xe^(-x)dx}*{∫（下限是0,上限是正无穷大）ye^(-y)dx}
={-∫（下限是0,上限是正无穷大）xde^(-x)}*{-∫（下限是0,上限是正无穷大）yde^(-y)}
={-xe^(-x)|(0,无穷）+∫（下限是0,上限是正无穷大）e^(-x)dx}*{-ye^(-y)|(0,无穷）+∫（下限是0,上限是正无穷大）e^(-y)dy}
={0-e^(-x)|（0,无穷）}*{0-e^(-y)|（0,无穷）}
={1}*{1}
=1
上面两个积分运算中用到了lim(x->无穷) xe^(-x)=0,这由络毕达法则容易得到.