Answer
$\displaystyle \frac{1}{2}\ln 5\approx 0.805$
check with desmos online calculator:
Work Step by Step
$I=\displaystyle \int_{-1}^{1}\frac{1}{2x+3}dx=$
Find the indefinite integral first,
$\displaystyle \int\frac{1}{2x+3}dx=\left[\begin{array}{ll}
u=2x+3 & \\
du=2dx & dx=\frac{1}{2}du
\end{array}\right]$
$=\displaystyle \frac{1}{2}\int\frac{1}{u}du=\frac{1}{2}\ln|u|+C$
$=\displaystyle \frac{1}{2}\ln|2x+3|+C$
Now, the definite integral:
$I=\left[\displaystyle \frac{1}{2}\ln|2x+3|\right]_{-1}^{1}$
$=\displaystyle \frac{1}{2}(\ln 5-\ln 1)$
$=\displaystyle \frac{1}{2}\ln 5$
$\approx 0.805$
