计算二重积分Y/sqr[(1+x^2+Y^2)^3],x和Y都是[0,1].

题目

计算二重积分Y/sqr[(1+x^2+Y^2)^3],x和Y都是[0,1].
答案是ln[(2+sqr2)/(1+sqr3)].若是算的辛苦,直接喊分数,我有的直接给.

答案

∫[0,1]∫[0,1]y/(1+x^2+y^2)^(3/2)dxdy
=∫[0,1]dx∫[0,1]y/(1+x^2+y^2)^(3/2)dy
=∫[0,1]dx∫[0,1]1/2(1+x^2+y^2)^(3/2)d(1+x^2+y^2)
=∫[0,1]dx 1/2*(-2)(1+x^2+y^2)^(-1/2) [0,1]
∫[0,1]{ (1+x^2)^(1/2)-(2+x^2)^(1/2)}dx
=ln(x+(1+x^2)^(1/2)-ln(x/√2+(1+1/2x^2)^(1/2)) [0,1]
=ln(1+√2)-ln(1/√2+√3/√2)
=ln(2+√2/1+√3)