Question

how was the problem of the integration of princely states solved after independence of India?

Answer 1

It was solved by the huge effort of Sardar Vallabh Bhai Patel. there were not many states left after independence, there were few like Kashmir and Hyderabad.

Answer 2

After the independence day the British left the choice to the princely states.
They can join with India or Pakistan or remain isolated.Then our freedom fighter Sardar Vallabhai Patel, the iron man of India went to those princely states with army forces and agreed them to join with India.