Question

关于数学积分定积分的一道题,求根号下(1-r^2)\(1+r^2)再乘以r在零到一的定积分

Answer 1

∫(0→1)√((1-r^2)/(1+r^2))*rdr=1/2∫(0→1)√(1-r^4)/(1+r^2)d(r^2)

令r^2=sint

则原式=1/2∫(0→π/2)cost/(1+sint)*costdt=1/2∫(0→π/2)(1-sin^2(t)/(1+sint)dt=1/2∫(0→π/2)(1-sint)dt=1/2t|(0→π/2)+1/2cost|(0→π/2)=π/4-1/2