Answer
\begin{aligned}
\int x^5 e^{x} d x
= e^{x}(x^5 -5x^4 +20x^3 -60x^2 +120x -120 )+C
\end{aligned}
Work Step by Step
Given $$\int x^{5} e^{x}d x $$
So, we have
\begin{array}{|c c c c|}\hline & Differentiation && {Integration} \\
\hline
+ \quad \rightarrow&x^{5} & & e^{x} \\
&&\searrow&\\ \hline
- \quad \rightarrow& 5x^4 & &e^{x} \\
&&\searrow&\\ \hline
+ \quad \rightarrow&20x^3& &e^{x} \\
&&\searrow&\\ \hline
- \quad \rightarrow&60x^{2} & & e^{x} \\
&&\searrow&\\ \hline
+ \quad \rightarrow&120x & & e^{x} \\
&&\searrow&\\ \hline
- \quad \rightarrow&120 & & e^{x} \\
&&\searrow&\\ \hline
&0 & &e^{x} \\
&&\\ \hline
\end{array}
Therefore
\begin{aligned}
I&=\int x^5 e^{x} d x\\
&=x^5e^{x}-5x^4e^{x}+20x^3e^{x}-60x^2e^{x}+120xe^{x}-120e^{x}\\
&= e^{x}(x^5 -5x^4 +20x^3 -60x^2 +120x -120 )+C
\end{aligned}
