Chapter 7 - Techniques of Integration - 7.1 Integration by Parts - 7.1 Exercises - Page 517: 52

$$\int x^ne^xdx = x^ne^x -n\int x^{n-1} e^xdx $$

Given $$\int x^ne^xdx$$ Let \begin{align*} u&=x^n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ dv=e^xdx\\ u&=nx^{n-1} dx\ \ \ \ \ \ \ \ \ \ \ v=e^x \end{align*} Then using integration by parts \begin{align*} \int x^ne^xdx&=uv-\int vdu\\ &= x^ne^x -n\int x^{n-1} e^xdx \end{align*}