3.3.8. Improper integral

Integrals with infinite limits and integrals of functions with infinite discontinuities on the integration interval are called improper integrals .

The improper integral of the function y = f ( x ) in the range from a to + ∞ is determined by the equality .

If this limit exists and is finite, then the improper integral converges; if the limit does not exist (or is equal to infinity), then the improper integral diverges .

Similarly, the following improper integrals are calculated:

If the function y = f ( x ) has an infinite discontinuity at the point and continuous for a ≤ x <c and c <x ≤ b , then .

The improper integral of a function that has an infinite discontinuity on the integration interval is called convergent if both limits exist: and divergent if at least one of these limits does not exist (or is equal to infinity).