Answer
$$2$$
Work Step by Step
$$\eqalign{
& \int_{ - 2}^2 {\frac{{{e^{z/2}}}}{{{e^{z/2}} + 1}}} dz \cr
& {\text{substitute }}u = {e^{z/2}} + 1,{\text{ }}du = \frac{1}{2}{e^{z/2}} \cr
& {\text{express the limits in terms of }}u \cr
& x = 2{\text{ implies }}u = {e^{2/2}} + 1 = e + 1 \cr
& x = - 2{\text{ implies }}u = {e^{ - 2/2}} + 1 = {e^{ - 1}} + 1 \cr
& {\text{the entire integration is carried out as follows}} \cr
& \int_{ - 2}^2 {\frac{{{e^{z/2}}}}{{{e^{z/2}} + 1}}} dz = 2\int_{{e^{ - 1}} + 1}^{e + 1} {\frac{1}{u}} du \cr
& {\text{find the antiderivative}} \cr
& = 2\left. {\ln u} \right|_{{e^{ - 1}} + 1}^{e + 1} \cr
& {\text{use the fundamental theorem}} \cr
& = 2\left( {\ln \left( {e + 1} \right) - \ln \left( {{e^{ - 1}} + 1} \right)} \right) \cr
& {\text{by logarithmic properties}} \cr
& = 2\ln \left( {\frac{{e + 1}}{{{e^{ - 1}} + 1}}} \right) \cr
& = 2\ln \left( {\frac{{e + 1}}{{\frac{{1 + e}}{e}}}} \right) \cr
& = 2\ln \left( e \right) \cr
& = 2 \cr} $$
