>>470
高校数学の範囲だ
∫[0,π/2]3sin(t)cos(t)√{a^2cos(t)^2+b^2sit(t)^2} dt
=3/(2√2)*∫[0,π/2]sin(2t)√{(a^2+b^2)+(a^2-b^2)cos(2t)} dt
a-b≠0の場合は
=-2^(-3/2){(a^2+b^2)+(a^2-b^2)cos(2t)}^(3/2)/(a^2-b^2)|_[t=0,π/2]
=-2^(-3/2){(2b^2)^(3/2)-(2a^2)^(3/2)}/(a^2-b^2)
=-(b^3-a^3)/(a^2-b^2)
=(a^2+ab+b^2)/(a+b)