integrate xe^xe^x-1^12

Let u = e^(x)-1, then du/dx = e^(x)

dx = du/e^(x)

The integral becomes:

∫(x*e^(x))/(e^(x)-1)^(1/2) dx

= ∫(x*e^(x))/(u)^(1/2) * (du/e^(x))

= ∫x*u^(-1/2) du

= 2*u^(1/2)*x - 2∫u^(1/2) dx

= 2√(e^x-1)x - 4/3(e^x-1)^(3/2) + C

where C is the constant of integration.