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: 49


$\approx 4.275$ check with desmos online calculator:

Work Step by Step

$I=\displaystyle \int_{0}^{4}\frac{5}{3x+1}dx=$ Find the indefinite integral first, $\displaystyle \int\frac{5}{3x+1}dx=5\int\frac{1}{3x+1}dx=\left[\begin{array}{ll} u=3x+1 & \\ du=3dx & dx=\frac{1}{3}du \end{array}\right]$ $=\displaystyle \frac{5}{3}\int\frac{1}{u}du=\frac{5}{3}\ln|u|+C$ $=\displaystyle \frac{5}{3}\ln|3x+1|+C$ Now, the definite integral: $I=\left[\displaystyle \frac{5}{3}\ln|3x+1|\right]_{0}^{4}$ $=\displaystyle \frac{5}{3}(\ln 13-\ln 1)$ $\approx 4.275$
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