Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.2 Exercises - Page 334: 45


$f(x)=-2\ln x+3x-2$

Work Step by Step

$f^{\prime}(x)=\displaystyle \int\frac{2}{x^{2}}dx=2\int x^{-2}dx\\=2\dfrac{x^{-1}}{-1}+C_{1}=-\dfrac{2}{x}+C_{1}$ since $f^{\prime}(1)=1,$ $1=-\displaystyle \frac{2}{1}+C_{1}$ $C_{1}=3\displaystyle \Rightarrow f^{\prime}(x)=-\frac{2}{x}+3$ $f(x)=\displaystyle \int(-\frac{2}{x}+3)dx=-2\int\frac{dx}{x}+3\int dx$ $f(x)=-2\ln|x|+3x+C$ since $f(1)=1, $and $x>0\Rightarrow|x|=x,$ $1=-2\ln 1+3\cdot 1+C$ $C=-2$ $f(x)=-2\ln x+3x-2$
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