令 x^(1/2) =t ,x = t^2 ,0≤t≤1
∫(0,1) xarctanx^(1/2)dx
= ∫(0,1) t^2 arctant d(t^2)
= 1/2 ∫(0,1) arctant d(t^4)
=[1/2 t^4 arctant ] (0,1) - 1/2 ∫(0,1) t^4 d(arctant)
=π/8 - 1/2 ∫(0,1) t^4/(1+t^2) dt
=π/8 - 1/2 ∫(0,1) (t^4-1+1)/(1+t^2) dt
=π/8 - 1/2 ∫(0,1) t^2 - 1 + 1/(1+t^2) dt
=π/8 -1/2 [1/3t^3 -t + arctant ] (0,1)
=π/8 - (-1/3+π/8)
=1/3