Answer
$$\frac{1}{2}\ln \left( {\frac{{11}}{5}} \right)$$
Work Step by Step
$$\eqalign{
& \int_0^{\ln 4} {\frac{{{e^x}}}{{3 + 2{e^x}}}dx} \cr
& {\text{substitute }}u = 3 + 2{e^x},{\text{ }}du = 2{e^x}dx \cr
& {\text{express the limits in terms of }}u \cr
& x = 0{\text{ implies }}u = 3 + 2{e^0} = 5 \cr
& x = \ln 4{\text{ implies }}u = 3 + 2{e^{\ln 4}} = 11 \cr
& {\text{the entire integration is carried out as follows}} \cr
& \int_0^{\ln 4} {\frac{{{e^x}}}{{3 + 2{e^x}}}dx} = \int_5^{11} {\frac{{1/2}}{u}} du \cr
& {\text{find the antiderivative}} \cr
& = \frac{1}{2}\left. {\left( {\ln \left| u \right|} \right)} \right|_5^{11} \cr
& {\text{use the fundamental theorem}} \cr
& = \frac{1}{2}\left( {\ln \left| {11} \right| - \ln \left| 5 \right|} \right) \cr
& {\text{Simplify}} \cr
& = \frac{1}{2}\ln \left( {\frac{{11}}{5}} \right) \cr} $$
