原式=∫dy∫dx/√(x²+y²) (改变积分次序)
=∫dθ∫rdr/r (作极坐标变换)
=∫dθ∫dr
=∫(sinθ/cos²θ)dθ
=-∫d(cosθ)/cos²θ
=-(-1)[1/cos(π/4)-1/cos0]
=√2-1.