To reach 1st jhana, do we have to let go, or do we have to put some effort? Recently, I've been reading a lot about the different methods and perspectives for reaching the jhanas.

For a long time, the only method I've been using is to "let go" everything, and to use the breath as a kind of anchor for the eventual and sporadic distraction. But I never felt anything like joy or rapture.

A few days ago, I found a meditation instruction where the instructor tells that effort and will can be used to give rise to joy and to expand it throughout the body. An as soon as I started doing that, I felt and important qualitative difference in the kind of feelings arising. 

Is this application of effort compatible with "letting go"?

What do the suttas tell us about this?

Kind regards!
The method you cite first is correct.  Do not pursue the second.  Any authentic feelings of joy and rapture will arise of their own accord.  Anything forced or willfully attained is not jhana.  The first stirrings of jhana will invariably catch you by surprise and will seem to come out of no where.  We cannot force them to arrive.    

If I had to guess, I'd say the main reason why you [and really most people] aren't experience jhana is that you haven't been sitting enough.  Outside of a retreat setting, daily 90 minute sits is really the bare minimum for effective samatha meditation.  Any less than that and the mind just isn't quelled enough for jhana factors to present themselves in any kind of stable way.  To be terribly honest, you are also highly unlikely to find full absorption outside of a retreat setting.  It usually takes at least three days of 10+ hours a day of sitting for the mind to release its hold on the world and the grosser manifestations of the hindrances.  Only afterwards is the mind prepared for jhana practice.