If [tex]\mathbf F[/tex] is conservative, then there is a scalar function [tex]f[/tex] such that
[tex]\nabla f(x,y)=\mathbf F(x,y)\iff\dfrac{\partial f}{\partial x}\,\mathbf i+\dfrac{\partial f}{\partial y}\,\mathbf j=e^{-y}\,\mathbf i-xe^{-y}\,\mathbf j[/tex]
Setting the first components equal to one another, we can integrate both sides to find
[tex]\dfrac{\partial f}{\partial x}=e^{-y}\implies f(x,y)=xe^{-y}+g(y)[/tex]
Differentiating both sides with respect to [tex]y[/tex] gives
[tex]\dfrac{\partial f}{\partial y}=-xe^{-y}+\dfrac{\mathrm dg}{\mathrm dy}=-xe^{-y}[/tex]
[tex]\implies\dfrac{\mathrm dg}{\mathrm dy}=0[/tex]
[tex]\implies g(y)=C[/tex]
so that
[tex]f(x,y)=xe^{-y}+C[/tex]
By the fundamental theorem of calculus, we have that
[tex]\displaystyle\int_{\mathcal C}\mathbf F\cdot\mathrm d\mathbf r=\int_{\mathcal C}\nabla f\cdot\mathrm d\mathbf r=f(1,1)-f(0,0)=\frac1e[/tex]
