x=a*sin(θ)とおくと、dx=a*cos(θ) dθより
∫[x=0〜a/2] {dx/(a^2-x^2)^(3/2) = (1/a^2)∫[x=0〜π/6] dθ/cos^2(θ)、{tan(θ)}'=1/cos^2(θ) だから、
(1/a^2){tan(θ)}|_[x=0〜π/6]=√3/(3a^2)