Current through inductor I know the derivation of formula that integral of voltage across an inductor gives the current through but I am unable to build an abstraction in my mind/ unable to physically picturize it in my mind that how can instantaneous sum of voltages result in instantaneous current
 A: This isn't how the physics works, but a useful mental model of a inductor in a circuit is that it adds inertia to the current.  Voltage pushes the current.  The current builds up according to how hard you push (the voltage) and how long you push (how long that voltage is applied).
Mathematically, that combination of how hard and how long you push is the time-integral of the voltage.
Note that there is a proportionality factor in there.  How much current exactly do you get for some voltage over some known time?  That's where the inductance comes in.  Just like more mass requires a harder or longer push to get to the same speed, more inductance requires a harder and longer push to get to the same current.  The proportionality factor to get current from the integral of the voltage is therefore the reciprocal of the inductance.
This mental model of inductance also helps to rationalize how a high voltage can be made with a inductor.  You build up current in the inductor, then open a switch to force the current to stop quickly.  Just like stopping a moving mass quickly, this generates a high force (voltage).  This is the basis for how a type of switching power supply called a boost converter works.  It repeatedly builds up current in a inductor, then tries to shut off that current quickly.  This makes little spurts of high voltage, which are captured and transferred to the output.
A: Here is how it goes and I hope I am able to satisfy your curiosity with my explanation. As soon as current passes through the coil, whose abstraction at the lumped circuit level is your inductor, it creates a magnetic field because a current through a coil creates a field through the enclosed area of the coil. This magnetic field was not there initially and hence this change of flux forces a reverse emf through the coil to balance the change in flux and create a reverse emf equal to applied potential at the ends of the coil. So, if the emf source is a time variable source, so is the current through the coil, so is the field through the centre of the coil, so is the flux through the coil, so is the back emf, so is the current. If you have any problem understanding it further, you could always put it in the comments section below.
A: It's easiest to understand if you take a derivative: the voltage is proportional to the change in current. This means that an inductor is like a bunch of people passing through a security checkpoint at a constant rate. The guards at this checkpoint are very short-sighted, they are OK with any constant rate of people going through the gate, they get acclimated to it. But when the rate seems to be decreasing, as if to keep their numbers up, they notice the decrease and start to grab and shove more people through the gates, until this balances out with the flow of people and they acclimate and no longer see the current changing. So they give some energy if it will get more people through, but only if the current is decreasing. Similarly if the constant rate starts to increase they will go forward and push on the people, slowing them down a bit, collecting their energy and storing it for later. They're trying to keep the flow constant by pushing the people, but they do not care what that constant is, and they store energy when the current is ramping up which they use to drive more current when it's ramping down.
In terms of voltages, the problem is a little more complicated. The complication is that a voltage difference can either come from an electric field, or from a changing vector potential. When the current is changing, the magnetic field in the inductor changes, and so the vector potential around the loop also is changing. 
