Prove that the lebesgue integrals $\lim_{n\to \infty} \int_{[0,1]^n} \frac{x_1^2+ \dots +x_n^2}{x_1+ \dots +x_n} \ d \mathrm{m}(\mathbf{x})=2/3$
Asymptotic behaviour of a multiple integral on the unit hypercube