Your first problem is that $F(x,y)= 2xe^y+ x^2e^y$ is NOT a "vector field". It is a scalar field. Did you mean $F(x,y)= 2xe^y\vec{ix}+ x^2e^y\vec{j}$?
Or is $\nabla F$ the integrand rather than F? And what is the path C? Is it a closed path?