Definition. Let f be a function which is continuous on the interval [a, b) but not at x = b. We define

If this limit exists and is finite then we say that the integral is convergent; otherwise, we say that the integral is divergent.

Definition. Let f be a function which is continuous on the interval (a, b] but not at x = a. We define

If this limit exists and is finite then we say that the integral is convergent; otherwise, we say that the integral is divergent.

Definition. Let f be a function which is continuous at all points x in the closed interval [a, b] except for some point c in (a, b). If both of

and

are convergent, then we define

and we say that the integral is convergent; otherwise, we say that the integral is divergent.