RECURSION TECHNIQUE.

Let $a$ be any fixed non$-$zero real number.

$F_0={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{ax}dx=\frac{1}{a}{e}^{ax}$

$F_1={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x{e}^{ax}dx=?$

$F_2={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^2{e}^{ax}dx=?$

etc etc....

In general $F_n={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^n{e}^{ax}dx$

We will now apply the integration by parts technique to $F_n.$ Your answers below should include expressions possibly using variables $n,a,x$ as well as exponential function base $e.$

$u=$ and $du=,dx$

$dv=,|dx$ and $v||=$

Then integration by parts shows

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^n{e}^{ax}dx=-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$

$=-,{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^{n-1}{e}^{ax}dx$