  Among the several types of closures of an ideal $I$ that have been defined
and studied in the past decades, the integral closure $\bar{I}$ has a central
place being one of the earliest and most relevant. Despite this role, it is
often a difficult challenge to describe it concretely once the generators of
$I$ are known. Our aim in this note is to show that in a broad class of ideals
their radicals play a fundamental role in testing for integral closedness, and
in case $I\neq \bar{I}$, $\surd{I}$ is still helpful in finding some fresh new
elements in $\bar{I}\setminus I$. Among the classes of ideals under
consideration are: complete intersection ideals of codimension two, generic
complete intersection ideals, and generically Gorenstein ideals.
