先化为极坐标
原式=∫(0,π/2)dθ∫(0,1) r√((1-r^2)/(1+r^2))dr
=π/2 * (1/2) ∫(0,1)√((1-r^2)/(1+r^2))dr^2
令t=r^2,则原式
=(π/4)∫(0,1)√((1-t)/(1+t))dt
因为分子分母都有根号,一般把分子有理化,把分母变成根号下二次
=(π/4)∫(0,1) (1-t)/√(1-t^2))dt
=(π/4)∫(0,1) d(arcsint + √(1-t^2))
=(π/4)(π/2 + 0 - 0 - 1)
=(π/8)(π-2)