The "undefined value" does test false?

sub if_integer{ 
    return defined $_[0] && $_[0] =~ m/^\d+$/; 
}

if_integer($_) and print $_, ' is an integer', $/ 
    for -1, 0, 1, 2, 2.2, 1E0, '1E2', 'A', '123', '123 fred'

0  is an integer
1  is an integer
2  is an integer
1  is an integer
123  is an integer
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So the logic of the test works fine.

Of course your regex is a little primitive as it excludes all negative integers, as well as those represent in exponential form, '1E0' for example.