First, think about how to answer your question:
Is 48 a multiple of 12?
How do you do this (without a computer)? You divide 48 by 12...
48 / 12 = 4
Okay, "4" has no decimals (e.g. no "." or "," depending on your locale preference), so the answer is yes.

Now let's do this in Perl:

# First make the program know the two numbers:
$Number1 = 48;
$Number2 = 12;

# Now divide the numbers:
$Result = $Number1 / $Number2;

# ...and look if we got a decimal:
[download]

Let's stop for a moment. There are many ways to do the check, here are some of them:

# Using your sub
is_integer_string($Result)

# Look for a .
$Result =~ /\./;

# Check if the number is the same as its integer part:
$Result == int($Result)
[download]

I prefer the last one. It compares the number to the integer part (e.g. everything before the . without rounding) and if they're the same - everything is fine.
Of cause, you need to add a

if (condition) { print "Yes"; } else { print "No"; }
[download]

Regarding your subroutine... I think it won't work as you expect: Leaving the \s* alone, you're matching a + sign followed by zero or one - sign followed by numbers. +3 matches, but the string representation of a number doesn't include a leading + in Perl. Better to put [] around the \+ and \- (and keep the ?). This will match either a -, a + or nothing before the number.
Perl could do a good part of the job for you if you want to match using a regexp:

sub is_integer_string {
 my $N = $_[0] + 0;
 return $N =~ /^\-?\d+$/;
}
[download]

Any mathematical operation will see the input variable as a number (including everything Perl accepts here). Adding 0 means: Don't change my number at all, but force it to be a Perl number with a known syntax which could be matched easily.

Update: Just noticed JavaFan's comment and he's right: modulo is the quickest way to do it, but all the others shown here also work.