Answer
$f(x,y,z)=e^{x} \sin yz+K$; Conservative
Work Step by Step
The vector field $F$ is conservative when $curl F=0$
When $F=ai+bj+ck$, then we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$
Now, $curl F=[(e^{x}\cos yz-yze^{x}\sin yz)-(e^{x} \cos yz-yze^{x} \sin yz)]i+[(ye^{x} \cos yz-ye^{x} \cos yz)]j+[(ze^{x} \cos yz-ze^{x} \cos yz)-k=0$
Thus, we have the vector field $F$ is conservative.
Consider $f(x,y,z)=e^{x} \sin yz+g(y,z) \implies f_y=z e^{x} \cos yz+g_y $ and So, $g'(y)=0$
Thus, we have $g_y=h(z)$ and $f_y=z (e^{x}) \cos (yz) $
Now, $f(x,y,z)=(e^{x}) \sin (yz)+h(z)$
Thus, $f_z=y (e^{x}) \cos (yz)+h'(z) \implies h'(z)=0$
Hence, we get $f(x,y,z)=e^{x} \sin yz+K$
