Answer
See proof
Work Step by Step
Certainly, here is your provided data written in LaTeX: Given the force field: \[ \mathbf{F}(x,y) = f(x,y) \mathbf{i} + g(x,y) \mathbf{j} \] where \[ \begin{align*} f(x,y) &= h(x)[x\sin y + y\cos y] \\ g(x,y) &= h(x)[x\cos y - y\sin y] \end{align*} \] If \(\mathbf{F}(x,y)\) is conservative, then \[ \frac{\partial f}{\partial x} = \frac{\partial g}{\partial y} \Rightarrow h(x)[x\cos y + \cos y - y\sin y] = h'(x)[x\cos y - y\sin y] + h(x)[\cos y] \Rightarrow h(x)[x\cos y - y\sin y] = h'(x)(x\cos y - y\sin y) \Rightarrow h(x) = h'(x) \Rightarrow h(x) = e^x +C \] where C is a constants. Taking $C$ = 0, $h(x) = e^x$, and \[ \mathbf{F}(x, y) = e^x (x \sin y + y \cos y) \mathbf{i} + e^x (x \cos y - y \sin y) \mathbf{j} \] is a conservative force field.
