Let v= 1-u, then s = ∫ (u+v+u 2 * v 2) (1/2) du = ∫ (u+v+u 2 * v 2) (65438+).

Multiply both sides of "=", and then

S^2=(∫∫(u+v+u^2*v^2)dudv)

=∫dv∫(u+v+u^2*v^2)du

=∫(u^2/2+uv+u^3/3*v^2+c 1)dv

=(u^2/2*v+uv^2/2+u^3*v^3/9+c 1v+c2)

√(u^2/2*v+uv^2/2+u^3*v^3/9+c 1v+c2)

= √[u^2/2*( 1-u)+u( 1-u)^2/2+u^3*( 1-u)^3/9+c 1*( 1-u)+c2]

PS: I have to admire the proof upstairs, a stroke of genius!