7.1 Integration By Parts/48

Prove $\int _{}^{}\left(x^{n}e^{x}\right)dx=x^{n}e^{x}-n\int _{}^{}\left(x^{n-1}e^{x}\right)dx$

$\int _{}^{}\left(x^{n}e^{x}\right)dx$

${\begin{aligned}u&=x^{n}\quad &dv=e^{x}dx\\[2ex]du&=nx^{n-1}dx\quad &v=e^{x}\\[2ex]\end{aligned}}$

${\begin{aligned}\int _{}^{}\left(x^{n}e^{x}\right)dx&=x^{n}e^{x}-\int _{}^{}\left(nx^{n-1}e^{x}\right)dx\\[2ex]&=x^{n}e^{x}-n\int _{}^{}\left(x^{n-1}e^{x}\right)dx\\[2ex]\end{aligned}}$