By Green's theorem, we have
[tex]\displaystyle\int_C\cos y\,\mathrm dx+x^2\sin y\,\mathrm dy=\iint_R\frac{\partial(x^2\sin y)}{\partial x}-\frac{\partial\cos y}{\partial y}\,\mathrm dx\,\mathrm dy[/tex]
where [tex]C[/tex] is the *boundary* of the rectangle [tex]R[/tex].
The integral is then
[tex]\displaystyle\int_0^5\int_0^2(2x+1)\sin y\,\mathrm dy\,\mathrm dx[/tex]
[tex]=\displaystyle(\cos0-\cos2)\int_0^5(2x+1)\,\mathrm dx[/tex]
[tex]=\displaystyle(1-\cos2)(5^2+5)=\boxed{30(1-\cos 2)}[/tex]
