Video: Will Smith Kicks the CRAP Out of Boxing Trainer

Will Smith beat the holy hell out of his kickboxing instructor recently -- crushing a training bag with MULTIPLE devastating shin strikes -- and TMZ has obtained the shockingly intense footage.

According to sources, Will is in the market for a new gym -- and had stopped by Brazen Boxing & MMA in Philadelphia earlier this week just to scope the place out.

Will stayed for a 20-minute MMA training session -- with badass Bellator fighter Jamall "X Factor" Johnson -- and we're told ... the actor's an EXTREMELY quick learner.