I am weak in such integral. May someone offer some method to solve it. Thanks $$\lim_{n\rightarrow\infty}\int_0^1 \int_0^1 \cdots\int_0^1 {n \over x_1+x_2+\cdots+x_n} \, dx_1 \, dx_2 \cdots dx_n$$
Sorry for don't give details. It's not a test from class or textbook. First I have no idea, then I see a similar problem which use the law of large numbers. But I don't know how to apply it.
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align}
&\lim_{n \to \infty}\int_{0}^{1}\cdots\int_{0}^{1}
{n \over x_{1} +  \cdots + x_{n}}\,\dd x_{1}\cdots\dd x_{n}
\\[5mm] = &
\lim_{n \to \infty}\braces{n\int_{0}^{1}\cdots\int_{0}^{1}
\bracks{\int_{0}^{\infty}\expo{-\pars{x_{1} +  \cdots + x_{n}}t}\dd t}
\dd x_{1}\cdots\dd x_{n}}
\\[5mm] = &\
\lim_{n \to \infty}\bracks{n\int_{0}^{\infty}\pars{\int_{0}^{1}\expo{-tx}
\,\dd x}^{n}\dd t} =
\lim_{n \to \infty}\bracks{n\int_{0}^{\infty}
\pars{1 - \expo{-t} \over t}^{n}\dd t}
\\[5mm] & =
\lim_{n \to \infty}\bracks{n\int_{0}^{\infty}
\exp\pars{n\,\ln\pars{1 - \expo{-t} \over t}}\dd t}
\\[5mm] & =
\lim_{n \to \infty}\bracks{n\int_{0}^{\infty}
\exp\pars{-\,{n \over 2}\,t}\dd t} = \bbx{2}
\end{align}

In the last two lines I used the
  Laplace Method.

