Answer
$A\approx 11.7686$
Work Step by Step
$A=\displaystyle \int_{a}^{b}f(x)dx$
$A=2\displaystyle \int_{1}^{4}xdx -\int_{1}^{4}\tan(0.3x)dx$
For the second integral,
$\left[\begin{array}{ll}
u=0.3x, & \\
du=0.3dx, & dx=\frac{1}{0.3}du=\frac{10}{3}du
\end{array}\right] $
Use the table on page 333.
$\displaystyle \int\tan(0.3x)dx=\frac{10}{3}\int\tan udu=\frac{10}{3}\ln|\cos u|+C$
$=\displaystyle \frac{10}{3}\ln|\cos 0.3x|+C$
$A=2\cdot\left[\frac{x^{2}}{2}\right]_{1}^{4}+\left[\frac{10}{3}\ln|\cos 0.3x|\right]_{1}^{4}$
$=16-1+\displaystyle \frac{10}{3}\ln|\cos 1.2x|-\frac{10}{3}\ln|\cos 0.3x|$
$A\approx 11.7686$
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