∫∫xe^xydxdy其中x大于等于0小于等于1y大于等于-1小于等于0求二重定积分

Question

∫∫xe^xydxdy 其中 x 大于等于0 小于等于1 y大于等于-1 小于等于0 求二重定积分

Answer 1

∫[0,1]∫[-1,0]xe^xydxdy
=∫[0,1]x∫[-1,0]e^xydydx
=∫[0,1]x*[lne^(xy)*e^(xy)][-1,0]dx
=∫[0,1]x*[(xy)*e^(xy)][-1,0]dx
=∫[0,1]x*[x*e^(-x)]dx
=∫[0,1]x^2*e^(-x)dx
=-x^2*e^(-x)[0,1]+∫[0,1]e^(-x)dx^2
=-1/e+2∫[0,1]xe^(-x)dx
=-1/e-2xe^(-x)[0,1]+2∫[0,1]e^(-x)dx
=-3/e+2∫[0,1]e^(-x)dx
=-3/e-2e^(-x)[0,1]
=2-5/e