Is the antiderivative of $\displaystyle \int_2^1{(1+(z/2))\,dx}$ is (z+(z^2/2))?
You wrote $\displaystyle dx$, implying that the integrand was to be integrated with respect to x, not z.
Also, you put in limits, so your final answer should be numerical, not algebraic.
Also, are you sure the limits are the right way around? Did you mean
$\displaystyle \int_1^2{1+\frac{z}{2}\,dz} $ ???
If so then:
$\displaystyle \int_1^2{1+\frac{z}{2}\,dz} $
$\displaystyle \int_1^2{1\,dz}+\int_1^2{\frac{z}{2}\,dz} $
$\displaystyle \int_1^2{1\,dz}+\frac{1}{2}\int_1^2{z\,dz} $
$\displaystyle [z]^2_1+\frac{1}{2}[\frac{z^2}{2}]^2_1$
$\displaystyle [2-1]+\frac{1}{2}[\frac{2^2}{2} - \frac{1^2}{2}]$
$\displaystyle 1+\frac{1}{2}[\frac{4}{2} - \frac{1}{2}]$
$\displaystyle 1+\frac{1}{2}[\frac{3}{2}]$
$\displaystyle 1+\frac{3}{4}$
$\displaystyle \frac{4}{4}+\frac{3}{4}$
$\displaystyle \frac{7}{4}$