I don't know whether to apply Stokes' Theorem or the Divergent Thereom to this problem I've been given!
"Calculate $\iint_S \vec F.d\vec S$, where $\vec F=x(x-1)yz\hat i+e^xsin(\pi y)\hat j +x^2y^2z\hat k$ and $S$ is the surface of the unit cube $0\le x,y,z\le 1$"
I suspect it's the Divergent Theorem (it's a closed surface because it's a cube, right?), but any help getting me started would be much appreciated!
$\nabla \cdot \vec F=(2x-1)yz+\pi e^x\cos (\pi y)+x^2y^2$
So, your taking the Divergence is correct.
Next, we see that the first and second terms integrate to zero.
It appears that you have the first integral correct.
For the second integral, recall that
$$\left.\int \cos (\pi y)dy=\frac{\sin \pi y}{\pi}\right|_0^1=0$$
For the integral of the third term, we have
$$\int_0^1\int_0^1\int_0^1x^2y^2dxdydz=1\times \frac13\times \frac13 =\frac19$$
Putting it all together, we have that the integral over the cube is
$$\bbox[5px,border:2px solid #C0A000]{\int_0^1\int_0^1\int_0^1((2x-1)yz+\pi e^x\cos (\pi y)+x^2y^2)dxdydz=\frac19}$$