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Calculus Level 5

⌊ ∫ 0 1 ∫ 0 1 ∫ 0 1 ∫ 0 1 ∫ 0 1 ⌊ x 1 + x 2 + x 3 + x 4 + x 5 ⌋ 2 d x 1 d x 2 d x 3 d x 4 d x 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 ∫ 0 1 … ∫ 0 1 f ( ⌊ x 1 + ⋯ + x n ⌋ ) d x 1 … d x n . \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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Kartik Sharma by Kartik Sharma , 21, India

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