∫x.e^x/ √(e^x-1)dx
= 2∫ xd√(e^x-1)
= 2x√(e^x-1) - 2 ∫ √(e^x-1)dx
let
e^(x/2) = seca
(1/2)e^(x/2) dx = (tana)^2 da
dx = 2(tana)^2/(seca) da
∫ √(e^x-1)dx
= ∫tana [2(tana)^2/(seca) ]da
=2 ∫ (sina)^3/(cosa)^2 da
= -2∫(1-(cosa)^2) / (cosa) ^2 dcosa
= 2 [ 1/cosa + cosa ] + C'
= 2[ e^(x/2) + e^(-x/2) ] + C'

∫x.e^x/ √(e^x-1)dx
= 2x√(e^x-1) - 2 ∫ √(e^x-1)dx
= 2x√(e^x-1) - 4 [ e^(x/2) + e^(-x/2) ]+ C