$\int_0^2 (|x| + |x - 2|) dx =$

The integral to evaluate is $\int_0^2 (|x| + |x - 2|) dx$

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For $|x|$ on the interval $[0, 2]$:

Since $x \geq 0$ throughout $[0, 2]$, we have $|x| = x$

For $|x - 2|$ on the interval $[0, 2]$:

Since $x \leq 2$ throughout $[0, 2]$, we have $x - 2 \leq 0$

Therefore $|x - 2| = -(x - 2) = 2 - x$

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Substituting into the expression:

$|x| + |x - 2| = x + (2 - x)$

$= x + 2 - x$

$= 2$

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The integral becomes:

$\int_0^2 2 \, dx$

$= 2x \Big|_0^2$

$= 2(2) - 2(0)$

$= 4$

Therefore, the value of the integral is $4$.