Answer
$$Area = 4 - \ln 5$$
Work Step by Step
$$\eqalign{
& y = \frac{x}{{1 + x}},{\text{ }}x{\text{ axis}}{\text{, line }}x = 4 \cr
& {\text{the area of the region is}} \cr
& Area = \int_0^4 {\frac{x}{{1 + x}}} dx \cr
& {\text{use long division}} \cr
& Area = \int_0^4 {\left( {1 - \frac{1}{{1 + x}}} \right)} dx \cr
& {\text{integrate}} \cr
& Area = \left. {\left( {x - \ln \left| {1 + x} \right|} \right)} \right|_0^4 \cr
& {\text{evaluate limits}} \cr
& Area = \left( {4 - \ln \left| {1 + 4} \right|} \right) - \left( {0 - \ln \left| {1 + 0} \right|} \right) \cr
& Area = 4 - \ln \left| {1 + 4} \right| - 0 + \ln \left| 1 \right| \cr
& Area = 4 - \ln 5 \cr} $$
