Answer
$5\ln 3$
Work Step by Step
$A=\displaystyle \int_{a}^{b}f(x)dx$
$A=\displaystyle \int_{1}^{5}\frac{5x}{x^{2}+2}dx=$
Find the indefinite integral...
$\displaystyle \int\frac{5x}{x^{2}+2}dx=\left[\begin{array}{ll}
u=x^{2}+2 & \\
du=2xdx & \frac{du}{2}=dx
\end{array}\right]=\displaystyle \frac{5}{2}\int\frac{du}{u}$
$=\displaystyle \frac{5}{2}\ln|u|+C=\frac{5}{2}\ln|x^{2}+2|+C$
$A=\displaystyle \frac{5}{2}[\ln|x^{2}+2|]_{1}^{5}=$
$A=\displaystyle \frac{5}{2}[\ln 27-\ln 3]=\frac{5}{2}\ln\frac{27}{3}$
$=\displaystyle \frac{5}{2}\ln 9=\frac{5}{2}\ln 3^{2}$
$=\displaystyle \frac{5}{2}\cdot 2\ln 3$
$=5\ln 3$
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