Answer
$z^{3}e^{z}-3z^{2}e^{z}+6ze^{z}-6e^{z}+C$
Work Step by Step
$\int$udv = uv - $\int$vdu
$\int$$z^{3}e^{z}$
u=$z^{3}$
du=3$z^{2}$dz
dv=$e^{z}$dz
v=$e^{z}$
$\int$$z^{3}e^{z}$ = $z^{3}$$e^{z}$ - $\int$$e^{z}$3$z^{2}$dz
= $z^{3}$$e^{z}$ - 3$\int$$e^{z}$$z^{2}$dz
Integration by parts again:
u=$z^{2}$
du=$2z$dz
dv=$e^{z}$dz
v=$e^{z}$
$z^{3}$$e^{z}$ - 3$\int$$e^{z}$$z^{2}$dz = $z^{3}$$e^{z}$ - 3($z^{2}$$e^{z}$ - $\int$$e^{z}$$2z$dz)
= $z^{3}$$e^{z}$ - 3($z^{2}$$e^{z}$ - 2$\int$$e^{z}$$z$dz)
Integration by parts one last time:
u=$z$
du=dz
dv=$e^{z}$dz
v=$e^{z}$
$z^{3}$$e^{z}$ - 3($z^{2}$$e^{z}$ - 2$\int$$e^{z}$$z$dz) = $z^{3}$$e^{z}$ - 3($z^{2}$$e^{z}$ - 2($z$$e^{z}$-$\int$$e^{z}$dz))
= $z^{3}$$e^{z}$ - 3($z^{2}$$e^{z}$ - 2($z$$e^{z}$-$e^{z}$))
= $z^{3}$$e^{z}$ - 3$z^{2}$$e^{z}$ - 6($z$$e^{z}$-$e^{z}$)
= $z^{3}e^{z}-3z^{2}e^{z}+6ze^{z}-6e^{z}+C$
