Answer
$\displaystyle \frac{15}{2}+8\ln 2$
Work Step by Step
$A=\displaystyle \int_{a}^{b}f(x)dx$
$A=\displaystyle \int_{1}^{4}\frac{x^{2}+4}{x}dx=\int_{1}^{4}\frac{x^{2}}{x}dx+\int_{1}^{4}\frac{4}{x}dx$
$=\displaystyle \int_{1}^{4}xdx+4\int_{1}^{4}\frac{1}{x}dx$
$=\left[\frac{x^{2}}{2}\right]_{1}^{4}+4\left[\ln x\right]_{1}^{4}$
$=\displaystyle \frac{4^{2}}{2}-\frac{1^{2}}{2}+4(\ln 4-\ln 1)$
$=\displaystyle \frac{15}{2}+4\ln 2^{2}$
$=\displaystyle \frac{15}{2}+8\ln 2$
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