Answer
$$2$$
Work Step by Step
$$\eqalign{
& \int_{\ln 4}^{\ln 9} {{e^{x/2}}} dx \cr
& {\text{use the formula }}\int_a^b {{e^{kx}}} dx = \left( {\frac{{{e^{kx}}}}{k}} \right)_a^b;{\text{ for this exercise set }}k = 1/2 \cr
& \int_{\ln 4}^{\ln 9} {{e^{x/2}}} dx = \left( {\frac{{{e^{x/2}}}}{{1/2}}} \right)_{\ln 4}^{\ln 9} \cr
& = 2\left( {{e^{x/2}}} \right)_{\ln 4}^{\ln 9} \cr
& {\text{use fundamental theorem of calculus }}\int_a^b {f\left( x \right)} dx = F\left( b \right) - F\left( a \right).\,\,\,\,\left( {{\text{see page 281}}} \right) \cr
& = 2\left( {{e^{\left( {ln9} \right)/2}} - {e^{\left( {ln4} \right)/2}}} \right) \cr
& {\text{simplifying}} \cr
& = 2\left( {{e^{\ln 3}} - {e^{\ln 2}}} \right) \cr
& = 2\left( {3 - 2} \right) \cr
& = 2 \cr} $$
