Vincent's NT Word Studies
8. Are willing (eudokoumen). The translation might well be made stronger as well as more literal: we are well-pleased.To be absent - present (ekdhmhsai - endhmhsai). The same verbs as in ver. vi. to be from home, at home.
Robertson's NT Word Studies
5:8 {We are of good courage} (qarroumen). Good word for cheer and same root as qarsew (#Mt 9:2,22). Cheer up. {Are willing rather} (eudokoumen). Rather, "We are well-pleased, we prefer" if left to ourselves. Cf. #Php 1:21f. Same eudokew used in #Lu 3:22. {To be at home with the Lord} (endemesai pros ton kurion). First aorist (ingressive) active infinitive, to attain that goal is bliss for Paul.