Example: Let f(x) = x2 and by power rule, f '(x) = 2x. Then the integral of f '(x) is, x2 + C, because by differentiating not only just x2 but also the functions such as x2 + 2, x2 - 1, etc gives 2x. The indefinite integral is techinically defined as shown below.
A definite integral represents a number when the lower and upper limits are constants. The indefinite integral represents a family of functions whose derivatives are f. The difference between any two functions in the family is a constant.