How to check if the binary representation of an integer is a palindrome?

Hopefully correct:

_Bool is_palindrome(unsigned n)
{
    unsigned m = 0;

    for(unsigned tmp = n; tmp; tmp >>= 1)
        m = (m << 1) | (tmp & 1);

    return m == n;
}

Since you haven't specified a language in which to do it, here's some C code (not the most efficient implementation, but it should illustrate the point):

/* flip n */
unsigned int flip(unsigned int n)
{
    int i, newInt = 0;
    for (i=0; i<WORDSIZE; ++i)
    {
        newInt += (n & 0x0001);
        newInt <<= 1;
        n >>= 1;
    }
    return newInt;
}

bool isPalindrome(int n)
{
    int flipped = flip(n);
    /* shift to remove trailing zeroes */
    while (!(flipped & 0x0001))
        flipped >>= 1;
    return n == flipped;
}

EDIT fixed for your 10001 thing.

Create a 256 lines chart containing a char and it's bit reversed char. given a 4 byte integer, take the first char, look it on the chart, compare the answer to the last char of the integer. if they differ it is not palindrome, if the are the same repeat with the middle chars. if they differ it is not palindrome else it is.