$\frac{{\int }_0^{\mathrm{\pi }\/3}{\int }_{\sin \left(x\right)}^{\sin \left(2 x\right)}F \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_{\mathrm{\pi }\/3}^{\mathrm{\pi }}{\int }_{\sin \left(2 x\right)}^{\sin \left(x\right)}F \mathit{\ⅆ}y \mathit{\ⅆ}x}{{\int }_0^{\mathrm{\pi }\/3}{\int }_{\sin \left(x\right)}^{\sin \left(2 x\right)}1 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_{\mathrm{\pi }\/3}^{\mathrm{\pi }}{\int }_{\sin \left(2 x\right)}^{\sin \left(x\right)}1 \mathit{\ⅆ}y \mathit{\ⅆ}x}$

$\=\frac{\frac{5}{2}\+4\mathrm{\pi }-\frac{15}{16}\sqrt{3}}{\frac{5}{2}}$

$\=1\+\frac{8}{5}\mathrm{\pi }-\frac{3}{8}\sqrt{3}$

$\doteq 5.377029193$

Solution from first principles

$F\=2 x\+3 y\+1$$\stackrel{\text{assign}}{\to }$

$\frac{{\int }_0^{\mathrm{\pi }\/3}{\int }_{\sin \left(x\right)}^{\sin \left(2 x\right)}F \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_{\mathrm{\pi }\/3}^{\mathrm{\pi }}{\int }_{\sin \left(2 x\right)}^{\sin \left(x\right)}F \mathit{\ⅆ}y \mathit{\ⅆ}x}{{\int }_0^{\mathrm{\pi }\/3}{\int }_{\sin \left(x\right)}^{\sin \left(2 x\right)}1 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_{\mathrm{\pi }\/3}^{\mathrm{\pi }}{\int }_{\sin \left(2 x\right)}^{\sin \left(x\right)}1 \mathit{\ⅆ}y \mathit{\ⅆ}x}$ = $1\+\frac{8}{5}\mathrm{\π}-\frac{3}{8}\sqrt{3}$$\stackrel{\text{at 10 digits}}{\to }$$5.377029193$$$$$

$F≔2 x\+3 y\+1\:$

$Q≔\frac{\mathrm{Int}\left(F\,\left[y\=\sin \left(x\right)..\sin \left(2 x\right)\,x\=0..\mathrm{\pi }\/3\right]\right)\+\mathrm{Int}\left(F\,\left[y\=\sin \left(2 x\right)..\sin \left(x\right)\,x\=\mathrm{\pi }\/3..\mathrm{\pi }\right]\right)}{\mathrm{Int}\left(1\,\left[y\=\sin \left(x\right)..\sin \left(2 x\right)\,x\=0..\mathrm{\pi }\/3\right]\right)\+\mathrm{Int}\left(1\,\left[y\=\sin \left(2 x\right)..\sin \left(x\right)\,x\=\mathrm{\pi }\/3..\mathrm{\pi }\right]\right)}$

$\frac{{\∫}_0^{\frac{1}{3}\mathrm{\π}}{\∫}_{\sin \left(x\right)}^{\sin \left(2x\right)}\left(2x\+3y\+1\right)\ⅆy\ⅆx\+{\∫}_{\frac{1}{3}\mathrm{\π}}^{\mathrm{\π}}{\∫}_{\sin \left(2x\right)}^{\sin \left(x\right)}\left(2x\+3y\+1\right)\ⅆy\ⅆx}{{\∫}_0^{\frac{1}{3}\mathrm{\π}}{\∫}_{\sin \left(x\right)}^{\sin \left(2x\right)}1\ⅆy\ⅆx\+{\∫}_{\frac{1}{3}\mathrm{\π}}^{\mathrm{\π}}{\∫}_{\sin \left(2x\right)}^{\sin \left(x\right)}1\ⅆy\ⅆx}$

$q≔\mathrm{value}\left(Q\right)$

$1\+\frac{8}{5}\mathrm{\π}-\frac{3}{8}\sqrt{3}$

$\mathrm{evalf}\left(q\right)$

$5.377029193$