Answer
$x^{3}e^{x}-3x^{2}e^{x}+6xe^{x}-6e^{x}+C$
Work Step by Step
$\int$$x^{3}e^{x}dx$
Use intergration by parts 3 times.
$u=x^{3}, dv=\int e^{x}dx$
$u'=3x^{2}dx,v=e^{x}$
$=x^{3}e^{x}-\int 3x^{2}e^{x}dx$
$u=-3x^{2},dv=\int e^{x}dx$
$u'=-6xdx,v=e^{x}$
$=x^{3}e^{x}-3x^{2}e^{x}+\int 6xe^{x}dx$
$u=6x,dv=\int e^{x}dx$
$u'=6dx,v=e^{x}$
$=x^{3}e^{x}-3x^{2}e^{x}+6xe^{x}-6e^{x}+C$
