Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 52

Use the formula $\displaystyle \int x^n e^x dx = x^n e^x - n \int x^{n-1} e^x dx$ to find $\displaystyle \int x^4 e^x dx$


$\begin{equation} \begin{aligned} \int x^4 e^x dx &= x^4 e^x - 4 \int x^3 e^x dx\\ \\ &= x^4 e^x - 4 \left( x^3 e^x - 3 \int x^2 e^x dx \right)\\ \\ &= x^4 e^x - 4x^3 e^x + 12 \left( x^2 e^x - 2 \int x e^x dx \right) \\ \\ &= x^4 e^x - 4x^3 e^x + 12 x^2 e^x - 24 \left( xe^x - \int x^0 e^x dx \right)\\ \\ &= x^4 e^x - 4x^3 e^x + 12x^2 e^x - 24 xe^x + 24 e^x + c\\ \\ &= e^x \left( x^4 - 4x^3 + 12 x^2 - 24 x + 24 \right) + c \end{aligned} \end{equation}$