A charge $q$ is placed at shown at a height $a$ above the center of a square surface of side length $a$. I have to find the flux due to $q$ through the surface.
I do not find any symmetric shape about the charge $q$. Had the heigh been $\frac a2$ I could have easily drawn a body centered cube and be done with it using Gauss Law and utilizing the symmetry.
Next I tried integrating but then I ended up with an integral like this $$\phi=\frac{q}{\pi \epsilon_0} \int_{0}^a \frac{1}{\sqrt{x^2+a^2}} \tan^{-1}\left({\frac{a}{\sqrt{x^2+a^2}}}\right)dx$$
On plugging it in here it says "Antiderivative or integral could not be found".
I'm sure I did the calculations right, and it was lengthy so I am not posting them here yet (though I will if I am asked to). I'm just curious that what does it mean when I cannot mathematically get the value of a flux even when I know that it physically exists, if the situation is replicated in real life? And if we need to find the value of the flux, what other tools can we use to do it?
