Do the protons inside the nucleus repel each other by the electrostatic force? If they do, why doesn't the repulsion drive the protons apart so that the nuclei get disintegrated?
There is an electrostatic repulsion between the protons in the nucleus. However, there is also an attraction due to another kind of force besides electromagnetism, namely the so-called "strong nuclear interaction".
The strong nuclear interaction ultimately boils down to the forces between the "colorful" quarks inside the protons - and neutrons. It is mediated by gluons, much like electromagnetism is mediated by photons, described by Quantum Chromodynamics (QCD), much like electromagnetism is described by Quantum Electrodynamics (QED), and it acts (almost) equally on protons and neutrons.
The attractive strong nuclear interaction inside the nuclei is 1-2 orders of magnitude stronger than the repulsive electrostatic interaction which is what keeps the nuclei together despite the repulsive electrostatic force.
Since protons are positively charged we think that they repel.....yes they repel due to electrostatic force between the protons(coulomb's law). In the similar fashion there also exists nuclear force ( nuclear binding energy) in the nucleus which balances the repulsion between the protons....so totally protons are balanced by both electrostatic force and nuclear force inside the nucleus...