Why the potential inside a solid conducting sphere is non zero while the electric field inside is zero? 
I don't know why V is not zero inside the sphere. If we use the formula that V= $\int_{0}^{\infty}(kdq/r)$, how do we get that $V_{in}$ = kq/r?  
 A: The potential difference inside a conductor is always zero [I edited your question]. The potential inside a conductor is not always zero. The potential is same at all points inside a conductor. Now, you see why the potential difference is zero. To obtain the expression for potential, you can use the expression,
$\int_{0}^{\infty}(kdq/r)$ in spherical polar coordinates. Express dq as $\rho$dV. Here dV is $r^2$sin($\theta$)drd$\theta$d$\phi$.

A: When you bring a test charge towards the sphere, you have to do some work on the charge to overcome the force force due to the electric field that is emerging from the sphere. This work will store itself in the test charge as it potential energy.
But precisely because the electric field inside the sphere is zero, you won't have to do any work. Thus the potential remains the same inside the sphere and equal to the potential of the charge at the outer boundary of the sphere.
You only have to do work till the outer boundary of the sphere. As long as there is movement of charge along(or against) the electric field, there will be work. No electric field means no work.
And the work that you have done till the outer boundary will appear as the potential energy of the charge inside the sphere. The charge inside the sphere still contains the potential energy that was stored in it when you did the work by bringing it from infinity to the outer boundary of the sphere.
FYI, potential means the work done by external agent per unit charge.
A: E is more or less change in potential so even if potential is constant then still E will be zero
Integral of dV from Vr to V = integral of  0 from R to x
And then we will get result as Vr is potential at surface which is found by continuity 
We have to take Vr as reference rather than Infinity 
A: We have to understand the difference of a vector quantity (electric field) and a scalar quantity (electric pontential). In a conducting solid or hollow sphere which is charged and that excess of free electrons, we know, are distribuited on the surface of the both spheres (solid or hollow) so there is no residual or net charge inside the sphere Qin=0 so Ein=0, so the electric field zero is the consequence of the charges (positive and negative) inside the conductor are neutralized and the charge excess (free electrons) migrate to the sphere surface. In relation to electric potential we have a scalar quantity that depends on the numerical value of r or it is taken into account of the difference of two points arbitrarily defined coz it does not depend on direction, it merely a summation of scalar values and in a long run, we calculate infinite integral limit and the diference results to the electric potential on the surface coz r is always equal to R that would be a straight line parallel to r or V=kQ/infinite-kQ/R      
