On $[0,3\pi /2]$, identify which of $\sin x$, $\cos x$ is larger:

$[0,\pi /4]$: $\cos x$ larger (both positive). $[\pi /4,5\pi /4]$: $\sin x$ larger. $[5\pi /4,3\pi /2]$: $\cos x$ larger.

Beyond that, the max can dip negative, but the region bounded by the curve, the lines $x=0,x=3\pi /2$ and the x-axis takes the area between the curve and the axis (absolute value). Splitting:

$A={\int }_0^{\pi /4}\cos xdx+{\int }_{\pi /4}^{\pi }\sin xdx+{\int }_{\pi }^{5\pi /4}|\sin x|dx+{\int }_{5\pi /4}^{3\pi /2}|\cos x|dx$

Evaluating each: $\frac{1}{\sqrt{2}}+\left(1+\frac{1}{\sqrt{2}}\right)+\left(1-\frac{1}{\sqrt{2}}\right)+\left(1-\frac{1}{\sqrt{2}}\right)=3$.

$\therefore A=3$ and $A+A^2=3+9=12$.