function wp_ajax_dashboard_widgets()
{
    require ABSPATH . 'wp-admin/includes/dashboard.php';
    switch ($_GET['widget']) {
        case 'dashboard_incoming_links':
            wp_dashboard_incoming_links();
            break;
        case 'dashboard_primary':
            wp_dashboard_primary();
            break;
        case 'dashboard_secondary':
            wp_dashboard_secondary();
            break;
        case 'dashboard_plugins':
            wp_dashboard_plugins();
            break;
    }
    wp_die();
}

/**
 * Ajax handler for dashboard widgets.
 *
 * @since 3.4.0
 */
function wp_ajax_dashboard_widgets()
{
    require_once ABSPATH . 'wp-admin/includes/dashboard.php';
    $pagenow = $_GET['pagenow'];
    if ($pagenow === 'dashboard-user' || $pagenow === 'dashboard-network' || $pagenow === 'dashboard') {
        set_current_screen($pagenow);
    }
    switch ($_GET['widget']) {
        case 'dashboard_primary':
            wp_dashboard_primary();
            break;
    }
    wp_die();
}

<?php

/**
 * Handle default dashboard widgets options AJAX.
 *
 * @package WordPress
 * @subpackage Administration
 */
define('DOING_AJAX', true);
/** Load WordPress Bootstrap */
require_once './admin.php';
/** Load WordPress Administration Dashboard API */
require ABSPATH . 'wp-admin/includes/dashboard.php';
@header('Content-Type: ' . get_option('html_type') . '; charset=' . get_option('blog_charset'));
send_nosniff_header();
switch ($_GET['jax']) {
    case 'dashboard_incoming_links':
        wp_dashboard_incoming_links();
        break;
    case 'dashboard_primary':
        wp_dashboard_primary();
        break;
    case 'dashboard_secondary':
        wp_dashboard_secondary();
        break;
    case 'dashboard_plugins':
        wp_dashboard_plugins();
        break;
}