Integral of a function is infinite when the area under the curve of the function is unbounded and extends infinitely at least in one direction.

This can happen when the function :

(a) has a vertical asymptote,

(b) has an infinite discontinuity within the interval of integration,

(c) itself is unbounded or approaches infinity within the interval of integration.

An infinite integral does not necessarily mean that the function itself is always infinite, but rather that the accumulated area under the curve over the interval of integration is infinite.