# Chapter 5 - Integration - 5.5 Substitution Rule - 5.5 Exercises: 52

$$\frac{1}{2}\ln \left( {\frac{{11}}{5}} \right)$$

#### 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}

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