The definitions of the three are different: the definition of generalized integral (generalized integral): generalized integral, also known as generalized integral, is a generalization of ordinary definite integral, which refers to an integral with infinite upper/lower limit or a defective integrand function. The former is called infinite generalized integral, and the latter is called loss integral (also called generalized integral of unbounded function).
Definition of deficient integral: deficient integral is a kind of calculus in higher mathematics, and it is a generalized integral with deficient points in the integrand function.
Definition of integral (specified integral): definite integral is a kind of integral, which is the integral of function f(x) and the limit in the interval [a, b].
The characteristics of the three are different: the characteristics of generalized integral (generalized integral): the integral interval is infinite. Characteristics of deficient integral: the function value is infinite at one point, but the area can be found.
Characteristics of constant integral (specified integral): If definite integral exists, it is a specific numerical value (area of curved trapezoid), while indefinite integral is a function expression, and they have only one mathematical relationship (Newton-Leibniz formula), and nothing else.
The nature of the three is different:
The nature of generalized integral (generalized integral): the upper and lower bounds are infinite, or the integrand has many defects, or the mixture of the above two is called mixed generalized integral. For mixed generalized integrals, it is necessary to split multiple integral intervals to make the original integral the sum of two separated infinite intervals and the generalized integrals of unbounded functions.
Properties of deficient integral: deficient integral is also called improper integral of unbounded function. The properties of constant integral (specified integral): 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.