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Calculus Level 5

\[\displaystyle \left \lfloor \int_0^1 { \int_0^1 { \int_0^1 { \int_0^1 {\int_0^1{{\left\lfloor x_1 + x_2 + x_3 + x_4 + x_5 \right \rfloor}^2 \, dx_1\, dx_2\, dx_3\, dx_4\, dx_5}}}}}\right \rfloor =\, ?\]


Bonus: Find a simplified general formula for \[\displaystyle \int_0^1{ \ldots \int_0^1{ f\big(\left\lfloor x_1 + \cdots + x_n\right\rfloor\big) dx_1 \ldots dx_n}}.\]

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