engage vs immerse

engage

verb
  • To engross or hold the attention of; to keep busy or occupied. 

  • To enter into conflict with (an enemy). 

  • To come into gear with. 

  • To enter into (an activity), to participate (construed with in). 

  • To draw into conversation. 

  • To mesh or interlock (of machinery, especially a clutch). 

  • To attract, to please; (archaic) to fascinate or win over (someone). 

  • To guarantee or promise (to do something). 

  • To arrange to employ or use (a worker, a space, etc.). 

  • To bind through legal or moral obligation (to do something, especially to marry) (usually in passive). 

  • To enter into battle. 

immerse

verb
  • To involve or engage deeply. 

  • To place within a fluid (generally a liquid, but also a gas). 

  • Archimedes determined the volume of objects by immersing them in water. 

  • To map into an immersion. 

How often have the words engage and immerse occurred in a corpus of books? (source: Google Ngram Viewer )