The definite integral is defined as the limit of a sum under two conditions (i) the interval [a, b] is of finite length and (ii) f is defined and bounded on [a, b]. Then is called a proper integral.
But there are many practical problems where f is unbounded on [a, b] or the interval is not finite. Such integrals are known as improper integrals.
For example: are improper integrals.