You need to evaluate the definite integral using the linearity of integral such that:

`int_(-x)^x (z - 1)/(z + 2) dz = int_(-x)^x z/(z + 2)dz - int_(-x)^x 1/(z+2) dz`

`int_(-x)^x z/(z + 2)dz = int_(-x)^x (z+2-2)/(z + 2)dz`

`int_(-x)^x z/(z + 2)dz = int_(-x)^x (z + 2)/(z + 2)dz - int_(-x)^x 2/(z + 2)dz`

`int_(-x)^x (z - 1)/(z + 2) dz = int_(-x)^x (z + 2)/(z + 2)dz - int_(-x)^x 2/(z + 2)dz - int_(-x)^x 1/(z+2) dz`

`int_(-x)^x (z - 1)/(z + 2) dz = int_(-x)^x dz - int_(-x)^x 3/(z + 2)` `dz`

` int_(-x)^x (z - 1)/(z + 2) dz = z|_(-x)^x - 3ln|z + 2||_(-x)^x`

`int_(-x)^x (z - 1)/(z + 2) dz = (x + x) - 3(ln|x+2| - ln|2-x|)`

`int_(-x)^x (z - 1)/(z + 2) dz = 2x - ln|(x+2)/(2-x)|^3`

**Hence, evaluating the definite integral yields `int_(-x)^x (z - 1)/(z + 2) dz = 2x - ln|(x+2)/(2-x)|^3.` **

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