Answer
$$4$$
Work Step by Step
$$\eqalign{
& \int_0^{\ln 16} {{e^{x/4}}} dx \cr
& {\text{use the formula }}\int_a^b {{e^{kx}}} dx = \left( {\frac{{{e^{kx}}}}{k}} \right)_a^b;{\text{ for this exercise set }}k = 1/4 \cr
& \int_0^{\ln 16} {{e^{x/4}}} dx = \left( {\frac{{{e^{x/4}}}}{{1/4}}} \right)_0^{\ln 16} \cr
& = 4\left( {{e^{x/4}}} \right)_0^{\ln 16} \cr
& {\text{use fundamental theorem of calculus }}\int_a^b {f\left( x \right)} dx = F\left( b \right) - F\left( a \right).\,\,\,\,\left( {{\text{see page 281}}} \right) \cr
& = 4\left( {{e^{\left( {\ln 16} \right)/4}}} \right) - 4\left( {{e^{0/4}}} \right) \cr
& = 4\left( {{e^{\left( {\ln {{16}^{1/4}}} \right)}}} \right) - 4\left( {{e^0}} \right) \cr
& {\text{simplifying}} \cr
& = 4\left( {{e^{\ln 2}}} \right) - 4\left( 1 \right) \cr
& = 4\left( 2 \right) - 4 \cr
& = 4 \cr} $$
