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It's easy.

You don't need jailbreak. In the native youtube app start a video and press the home button. Then double tap the home button, swipe the multitasking bar to the right and press play. It's easy. Then you can do whatever you like with the music in the background.
 
You don't need jailbreak. In the native youtube app start a video and press the home button. Then double tap the home button, swipe the multitasking bar to the right and press play. It's easy. Then you can do whatever you like with the music in the background.

When this question was first asked, iOS 4 was unable to do it that way. Thus, the need to jailbreak.
 
This still doesn't work. How do you open a youtube video in Safari/or any other browser and let the youtube player play it while keep continue browsing?
 
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