[tex]\vec f(x,y,z)=(2yze^{2xyz}+4z^2\cos(xz^2))\,\vec\imath+2xze^{2xyz}\,\vec\jmath+(2xye^{2xyz}+8xz\cos(xz^2))\,\vec k[/tex]
Let
[tex]\vec f=f_1\,\vec\imath+f_2\,\vec\jmath+f_3\,\vec k[/tex]
The curl is
[tex]\nabla\cdot\vec f=(\partial_x\,\vec\imath+\partial_y\,\vec\jmath+\partial_z\,\vec k)\times(f_1\,\vec\imath+f_2\,\vec\jmath+f_3\,\vec k)[/tex]
where [tex]\partial_\xi[/tex] denotes the partial derivative operator with respect to [tex]\xi[/tex]. Recall that
[tex]\vec\imath\times\vec\jmath=\vec k[/tex]
[tex]\vec\jmath\times\vec k=\vec i[/tex]
[tex]\vec k\times\vec\imath=\vec\jmath[/tex]
and that for any two vectors [tex]\vec a[/tex] and [tex]\vec b[/tex], [tex]\vec a\times\vec b=-\vec b\times\vec a[/tex], and [tex]\vec a\times\vec a=\vec0[/tex].
The cross product reduces to
[tex]\nabla\times\vec f=(\partial_yf_3-\partial_zf_2)\,\vec\imath+(\partial_xf_3-\partial_zf_1)\,\vec\jmath+(\partial_xf_2-\partial_yf_1)\,\vec k[/tex]
When you compute the partial derivatives, you'll find that all the components reduce to 0 and
[tex]\nabla\times\vec f=\vec0[/tex]
which means [tex]\vec f[/tex] is indeed conservative and we can find [tex]f[/tex].
Integrate both sides of
[tex]\dfrac{\partial f}{\partial y}=2xze^{2xyz}[/tex]
with respect to [tex]y[/tex] and
[tex]\implies f(x,y,z)=e^{2xyz}+g(x,z)[/tex]
Differentiate both sides with respect to [tex]x[/tex] and
[tex]\dfrac{\partial f}{\partial x}=\dfrac{\partial(e^{2xyz})}{\partial x}+\dfrac{\partial g}{\partial x}[/tex]
[tex]2yze^{2xyz}+4z^2\cos(xz^2)=2yze^{2xyz}+\dfrac{\partial g}{\partial x}[/tex]
[tex]4z^2\cos(xz^2)=\dfrac{\partial g}{\partial x}[/tex]
[tex]\implies g(x,z)=4\sin(xz^2)+h(z)[/tex]
Now
[tex]f(x,y,z)=e^{2xyz}+4\sin(xz^2)+h(z)[/tex]
and differentiating with respect to [tex]z[/tex] gives
[tex]\dfrac{\partial f}{\partial z}=\dfrac{\partial(e^{2xyz}+4\sin(xz^2))}{\partial z}+\dfrac{\mathrm dh}{\mathrm dz}[/tex]
[tex]2xye^{2xyz}+8xz\cos(xz^2)=2xye^{2xyz}+8xz\cos(xz^2)+\dfrac{\mathrm dh}{\mathrm dz}[/tex]
[tex]\dfrac{\mathrm dh}{\mathrm dz}=0[/tex]
[tex]\implies h(z)=C[/tex]
for some constant [tex]C[/tex]. So
[tex]f(x,y,z)=e^{2xyz}+4\sin(xz^2)+C[/tex]
