\[\int _{ 0 }^{ \infty }{ \left\lfloor \frac { 6 }{ { e }^{ x } } \right\rfloor \, dx } =\ln { \frac { { A }^{ B } }{ C! } } \]

The above equation is true for positive integers \(A\), \(B\) and \(C\).

Find the minimum value of \(A+B+C\).

\[\]

**Bonus:** Find the generalization of \(\displaystyle \int _0^\infty \left \lfloor \frac n{e^x} \right \rfloor \, dx \).

**Notation**: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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