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  4. ∫ limits0∞ xn e-x d x (n is a+v e integer) is equal to

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Q. $\int\limits_0^{\infty} x^n e^{-x} d x \,(n$ is $a+v e$ integer) is equal to

Integrals

Solution:

$I_n=\left(-x^n e^{-x}\right)_0^{\infty}+n \int\limits_0^{\infty} x^{n-1} e^{-x} d x=0+n I_{n-1}=\ldots \ldots . .$
$=n !=\int\limits_0^{\infty} e^{-x} d x=n !$