>>649
∬_D {(a^2+x^2+y^2)^(-3/2)}dxdy
=∫[0,a]dy [ x(a^2+x^2+y^2)^(-1/2) * (a^2+y^2)^(-1)][x=0,a]
=a∫[0,a] {(2a^2+y^2)^(-1/2) * (a^2+y^2)^(-1)}dy
t=y+√(y^2+2a^2) とおいて置換
=a∫[(√2)a,(1+√3)a] {4t/(t^4+4a^4)}dt
u=t^2/(2a^2) と置換
=(1/a)∫[1,2+√3] {1/(u^2+1)}du
=(1/a){arctan(2+√3)-π/4}