## Calculus: Early Transcendentals (2nd Edition)

Published by Pearson

# Chapter 7 - Integration Techniques - 7.5 Partial Fractions - 7.5 Exercises - Page 550: 52

#### 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}

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