Question

$\underset{0}{\overset{\infty }{\int }}{x}^n{e}^{-x}dx(n$ is $a+ve$ integer) is equal to

• A. $n!$
• B. $(n-1)!$
• C. $(n-2)$ !
• D. $(n+1)!$

Answer 1

Answer: (A)

$I_n={\left(-x^n{e}^{-x}\right)}_0^{\infty }+n\underset{0}{\overset{\infty }{\int }}{x}^{n-1}{e}^{-x}dx=0+n{I}_{n-1}=\dots \dots ..$
$=n!=\underset{0}{\overset{\infty }{\int }}{e}^{-x}dx=n!$