the reason why u have 'C' is that the original function could have a constant added to it
eg. y = x+1, y' = 1
If u integrate back, u get:
y = x + C
eg. y = x, y' = 1
If u integrate back, u get:
y = x + C
C is determined by a relation between x and y
Another example, y' = x^2 /2
Integrate...