Chapter 7.3, Problem 21E

To determine

To determine the integral ${\displaystyle \int x^4{e}^{-x}dx}$

Answer to Problem 21E

The value of the integral ${\displaystyle \int x^4{e}^{-x}dx}$ is $e^{-x}\left[-x^4-4x^3-12x^2-24x-24\right]$

Explanation of Solution

Given information: ${\displaystyle \int x^4{e}^{-x}dx}$

Formula used: ${\displaystyle \int uvdx=u{\displaystyle \int vdx-{\displaystyle \int \left(u^{\prime }{\displaystyle \int vdx}\right)}}}dx$

Calculation :

Differentiation of $u(x)$	Signs	Integration of $v(x)$
$x^4$	(+)	$e^{-x}$
$4x^3$	(-)	$-e^{-x}$
$12x^2$	(+)	$e^{-x}$
$24x$	(-)	$-e^{-x}$
$24$	(+)	$e^{-x}$
0	(-)	$-e^{-x}$

  ${\displaystyle \int x^4{e}^{-x}dx}=(x^4)(-e^{-x})-(4x^3)(e^{-x})+(12x^2)(-e^{-x})-(24x)(e^{-x})+(24)(-e^{-x})$

Hence, ${\displaystyle \int x^4{e}^{-x}dx}=e^{-x}\left[-x^4-4x^3-12x^2-24x-24\right]$