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$\int_0^4|| x-2|-x| d x=$
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The correct answer is:
6
Given $\int_0^4|| x-2|-x| d x$
Let $I=\int_0^4|| x-2|-x| d x$
$\begin{aligned}
& \Rightarrow \mathrm{I}=\int_0^2|2-2 \mathrm{x}| \mathrm{dx}+\int_2^4 2 \mathrm{dx} \\
& =2 \int_0^1(1-\mathrm{x}) \mathrm{dx}+2 \int_1^2(\mathrm{x}-1) \mathrm{dx}+2[4-2] \\
& =1+1+4=6
\end{aligned}$
Let $I=\int_0^4|| x-2|-x| d x$
$\begin{aligned}
& \Rightarrow \mathrm{I}=\int_0^2|2-2 \mathrm{x}| \mathrm{dx}+\int_2^4 2 \mathrm{dx} \\
& =2 \int_0^1(1-\mathrm{x}) \mathrm{dx}+2 \int_1^2(\mathrm{x}-1) \mathrm{dx}+2[4-2] \\
& =1+1+4=6
\end{aligned}$
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