The value range of integral variable of definite integral can be an open interval or a closed interval. Because the definite integral is to find the area surrounded by the image of the function f(X) in the interval [a, b]. That is, the area of the graph surrounded by y = 0, x = a, x = b and y = f (x). The value obtained after integration is definite, constant, not a function.

A function can have indefinite integral, but not definite integral; There can also be definite integral, but there is no indefinite integral. A continuous function must have definite integral and indefinite integral; If there are only a finite number of discontinuous points, the definite integral exists; If there is jump discontinuity, the original function must not exist, that is, the indefinite integral must not exist.

Extended data:

The related theorem of definite integral;

Theorem 1: If f(x) is continuous in the interval [a, b], then f(x) is integrable in [a, b].

Theorem 2: If the interval f(x) is bounded on [a, b] and there are only finite discontinuous points, then f(x) is integrable on [a, b].

Theorem 3: Let f(x) be monotone in the interval [a, b], then f(x) can be integrated in [a, b].