By Green's theorem,
[tex]\displaystyle\int_C(3y+5e^x)\,\mathrm dx+(8x+9\cos(y^2))\,\mathrm dy[/tex]
[tex]\displaystyle=\iint_D\left(\frac{\partial(8x+9\cos(y^2))}{\partial x}-\frac{\partial(3y+5e^x)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=5\iint_D\mathrm dx\,\mathrm dy[/tex]
where [tex]D[/tex] is the region enclosed by [tex]C[/tex]. Equivalently, the line integral is equal to 5 times the area of [tex]D[/tex], which is
[tex]\displaystyle5\iint_D\mathrm dx\,\mathrm dy=5\int_0^1\int_{x^2}^{\sqrt x}\mathrm dy\,\mathrm dx=5\int_0^1(\sqrt x-x^2)\,\mathrm dx=\boxed{\frac53}[/tex]
