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\begin{aligned}\displaystyle \int_0^1 \cdots \int_{0}^1 \left[\sum_{i=1}^n x_i\right] dx_1\cdots dx_n &=\int_0^1 \cdots \int_{0}^1 \left[n-\sum_{i] dx_1\cdots dx_n\\&=\int_0^1 \cdots \int_{0}^1 \frac{n-1}{2}\;dx_1\cdots dx_n\\&=\frac{n-1}{2}\end{aligned}
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