Hot Integral - 28

Calculus Level 5

\[\displaystyle \lim_{n \to \infty}\frac{1}{\Gamma(n+1)} \int\limits_{0}^{\infty} \int\limits_{0}^{\infty} \int\limits_{0}^{\infty} \dots \int\limits_{0}^{\infty} \frac{1}{(\prod_{i=1}^{n} x_i)(\sum_{i=1}^{n} x_i +\frac{1}{x_i})^2}\, dx_1 dx_2 \cdots dx_n \]

The above expression can be expressed as \[\large \frac{A}{Be^{C\gamma}}\]

Write the answer as the concatenation of the sum \(ABC+CBA\) of the integers \(A,B,C\). For example, if you think that \(A=1,B=2,C=3\), type the answer as \(123+321=\boxed{444}\).

Moreover , if you think that the above expression is not possible and you think that limit does not exist, enter your answer as 888.

This is a part of Hot Integrals

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