..Find the area enclosed by the x-axis and the curve x = 1+e^t, y = t-t^2

My work:

ln(x-1) = t

y = ln (x-1) - (ln (x-1))^2

Area = Integral from 2 to e+1 of (t-t^2)(e^t) dt x goes from 2 to e+1.

After integrating I get e^t(1+t-t^2) from 0 to e+1 But t goes from 0 to1.

Which turns out to be something like -10. What am I doing wrong?