Hi Viper,
I included reCaptcha in my simple component by setting up reCaptch in the usual contact form and then included the following code to bring it in. Possibly not the "right" or maybe "best" way but it worked for me.
In your front-end model XML file include;
<field
name="captcha"
type="captcha"
label="COM_USERS_CAPTCHA_LABEL"
description="COM_USERS_CAPTCHA_DESC"
validate="captcha"
/>
And then in your default.php view place the following where you want reCaptcha to show up.
<div class="control-group">
<div class="control-label"><?php echo $this->form->getLabel('captcha'); ?></div>
<div class="controls"><?php echo $this->form->getInput('captcha'); ?></div>
</div>
I think this is all I did..... Good luck.
Glenn