I agree with SpiceMan (but will write in english, pardon).
Shin you : to trust, to believe, "credit" (as in latin "credere", spanish creer), from there Shin you ginko etc.
Shin rai : to have faith (in) , to believe (but religious sense)
I would think that Flam's proposed :"shinrai shite moraitai/kudasai" would mean "please,[I want you to] have faith in me" , a bit stronger than the original I want you to trust me which could well be :shinyou shite kureru/kuremasuka (moraitai/kudasai).