Answer
$$x^{3} e^{x}-3 x^{2} e^{x}+6 x e^{x}-6 e^{x}+C$$
Work Step by Step
Given $$\int x^3 e^x dx$$
Use
\begin{align*}
\int x^n e^x dx&= x^n e^x -n\int x^{n-1} e^xdx
\end{align*}
Then for $n=3$
\begin{align*}
\int x^3 e^x dx&= x^3 e^x -3\int x^{2} e^xdx \\
&= x^3 e^x -3\left( x^2e^x -2\int x e^xdx \right) \\
&= x^3 e^x -3 x^2e^x +6\int x e^xdx\\
&= x^3 e^x -3 x^2e^x +6 \left(x e^x-\int e^xdx\right)\\
&= x^{3} e^{x}-3 x^{2} e^{x}+6 x e^{x}-6 e^{x}+C
\end{align*}
