Let $X \sim \exp(\lambda)$. Find the Moment generating function of $Y=1+\ln(X)$.
I tried this but got myself stuck in an infinite descent of Integration by Parts. This makes me think there is an easier method.
$$m_Y(t) = E[e^{t(1+\ln(x)}] = \int_0^\infty e^{t(1+\ln(x)} \lambda e^{-\lambda x}dx = \lambda e^{t} \int_0^\infty x^te^{-\lambda x} dx$$
The issue is that when integrating by parts I try to reduce $x^t$ and then $x^{t-1}$, and I do not see a stopping point.