I need some help integrating :
$\displaystyle \int e^{-|y|}e^{ty} dy $
You just have to split it up as a sum of two integrals, by considering saparately the intervals on which y is positive and negative:
$\displaystyle \int e^{-|y|}e^{ty}\, dy = \int_{y\leqslant0} e^{y}e^{ty}\, dy + \int_{y\geqslant0} e^{-y}e^{ty}\, dy = \int_{y\leqslant0} e^{(t+1)y}\, dy + \int_{y\geqslant0} e^{(t-1)y}\, dy$