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If $a \in \mathbb{Z}^{+},[x]$ is greatest integer not more than $x$ and $\int_0^a 2^{[x]} d x=127$, then $a=$
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$8$
$\begin{aligned} & \text {} \int_0^a 2^{[x]} d x \\ & =\int_0^1 2^0 d x+\int_1^2 2^1 d x+\int_2^3 2^2 d x+\ldots \int_{a-1}^a 2^{a-1} d x=127 \\ & \Rightarrow 2^0+2^1+2^2+\ldots+2^{a-1}=127 \\ & \Rightarrow \frac{1\left[2^a-1\right]}{2-1}=127 \Rightarrow 2^a=128 \Rightarrow 2^a=2^8 \Rightarrow a=8\end{aligned}$
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