How do you record your iPhone's screen?

I would like to be able to record my iPhone's screen (for demos and stuff like that). However, there are some limitations:

1. No jailbreak—I am not jailbroken and am completely uninterested in becoming so.

2. $0 to spend—the solution must be free.

3. Nothing crazily complicated—though I am capable of doing advanced things with computers, if I mess up our computer it would not be good.

4. Downloads must be safe and minimal—in our house we are very worried about privacy and hackers so there should really only be one download involved. Also, my iPhone is only 16GB, so the download can't be too big. Also, this is my main and only phone and our main and only computer we are talking about. This can't be one of those things where they say "oh, it might crash your phone." This has to be a tried, working, safe and reliable solution.

I know it's a tough question, but I'm hoping someone somewhere will have an answer.

Thanks! :)