[tex]\mathbf f(x,y)=(ye^x+\sin y)\,\mathbf i+(e^x+x\cos y)\,\mathbf j[/tex] is continuous, so [tex]\mathbf f(x,y)[/tex] is indeed conservative. So we know there is some scalar function [tex]f(x,y)[/tex] such that [tex]\nabla f(x,y)=\mathbf f(x,y)[/tex].
[tex]\dfrac{\partial f}{\partial x}=ye^x+\sin y[/tex]
[tex]\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=\int(ye^x+\sin y)\,\mathrm dx[/tex]
[tex]f(x,y)=ye^x+x\sin y+g(y)[/tex]
[tex]\dfrac{\partial f}{\partial y}=\dfrac{\partial(ye^x+x\sin y+g(y))}{\partial y}[/tex]
[tex]e^x+x\cos y=e^x+x\cos y+g'(y)[/tex]
[tex]0=g'(y)[/tex]
[tex]\implies g(y)=C[/tex]
So we have
[tex]f(x,y)=ye^x+x\sin y+C[/tex]
