The value of

MATHEMATICS · Single correct · AoPS · Mathematics · Special Functions · Special Functions

Problem

The value of ∫0x[t+1]3 dt\int_0^x[t+1]^3\,dt, where [ ⋅ ][\,\cdot\,] denotes the greatest integer function, is equal to (1) ([x]([x]+1)2)2+([x]+1)3{x}\left(\frac{[x]([x]+1)}{2}\right)^2+([x]+1)^3\{x\} (2) ([x]([x]+1)2)3+([x]+1)3{x}\left(\frac{[x]([x]+1)}{2}\right)^3+([x]+1)^3\{x\} (3) ([x]([x]+1)2)3+([x]+1)2{x}\left(\frac{[x]([x]+1)}{2}\right)^3+([x]+1)^2\{x\} (4) None of these

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