/**
1)Get recaptcha keys from http://www.google.com/recaptcha

2)Set these keys to recaptcha plugin and activate it if it's not.

3)Put below code where you want to show recaptcha
*/
//php code
JPluginHelper::importPlugin('captcha');
$dispatcher = JDispatcher::getInstance();
$dispatcher->trigger('onInit','dynamic_recaptcha_1');

//html code inside form tag
<div id="dynamic_recaptcha_1"></div>
//4)Put this code where you validating/processing the form

$post = JRequest::get('post');      
JPluginHelper::importPlugin('captcha');
$dispatcher = JDispatcher::getInstance();
$res = $dispatcher->trigger('onCheckAnswer',$post['recaptcha_response_field']);
if(!$res[0]){
    die('Invalid Captcha');
}