>700
　1/(e^x -1) = e^(-x)/{1-e^(-x)} = Σ[k=1,∞) e^(-kx)
　∫_[0,a] (x^n)e^(-kx)dx = {1/[k^(n+1)]}∫_[0,ka] (y^n)e^(-y)dy
　∫(y^n)e^(-y)dy = -{y^n + n･y^(n-1) + … + (n!)y + (n!)}e^(-y) + c.　(←部分積分n回)
　∴ ∫_[0,∞) (y^n)e^(-y)dy = (n!)∫_[0,∞) e^(-y)dy = n!
　∴ I_n ≡∫_[0,∞) (x^n)/(e^x -1) dx = n! Σ[k=1,∞) 1/[k^(n+1)] = n!ζ(n+1).
　n=3 のとき、I_3 = 6× (π^4)/90 = = 6.49393940226683…