Question

Derive the following reduction formula :

Integral x^n e^x dx=x^n e^x-n integral x^n-1 e^x dx

Answer 1

Given

$\int x^n e^xdx$

Let

$\begin{aligned} u&=x^n \ \ \ \ \ \ \ &dv=&e^xdx\\ du&=nx^{n-1}dx \ \ \ \ \ \ \ &v=&e^x \end{aligned}$

Then

$\begin{aligned} \int x^n e^xdx&= uv-\int vdu\\ &=x^n e^x- n \int x^{n-1} e^xdx \end{aligned}$