Answer
$\approx 4.275$
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:
$I=\left[\displaystyle \frac{5}{3}\ln|3x+1|\right]_{0}^{4}$
$=\displaystyle \frac{5}{3}(\ln 13-\ln 1)$
$\approx 4.275$
