∫∫√1-x^2-y^2/1+x^2+y^2dxdy,其中D为区域x^2+y^2≤1的二重积分计算

原式= ∫dθ∫√(1-r²)/(1+r²)rdr (极坐标变换)
=π∫√(1-r²)/(1+r²)d(r²)
=2π∫t²dt/(2-t²) (令√(1-r²)=t)
=√2π∫[1/(√2+t)+1/(√2-t)-√2]dt
=√2π[ln│√2+t│-ln│√2-t│-√2t]│
=√2π[ln│(√2+t)/(√2-t)│-√2t]│
=√2π[ln│(√2+1)/(√2-1)│-√2]
=√2π[2ln(√2+1)-√2]
=2π[√2ln(√2+1)-1]。

原式= ∫[0,2π] dθ ∫[0,1] √(1-r²)/(1+r²) r dr (极坐标变换) = π ∫[0,1]√(1-r²)/(1+r²)d(r²) 令 u= r² = π ∫[0,1] √(1-u) / √(1+u) du = π ∫[0,...