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An indefinite integral is the set of all antiderivatives some function
A definite integral of the function f (x) on the interval [a; b] is the limit of integral sums when the diameter of the partitioning tends to zero if it exists independently of the partition and choice of points inside the elementary segments.
Proper integral is a definite integral, which is bounded as expanded function, and the region of integration.
Improper integral is definite integral, which is unlimited or expanded function, or the region of integration, or both together
Then function defined on the half-line and integrable on any interval The limit of the integral and is called the improper integral of the first kind of function a to and