Example:
\[I= \int^{\infty}_0 x^ne^{-x} dx\]
.Let
\[I_n= \int^{\infty}_0 x^ne^{-x} dx\]
. Integration by parts gives\[\begin{equation} \begin{aligned} I_n &= \int^{\infty}_0 x^ne^{-x} dx \\ &= [-nx^{n-1} e^{-x} ]^{\infty}_0 -(-n \int^{\infty}_0 x^{n-1}e^{-x} dx \\ &= n \int^{\infty}_0 x^{n-1}e^{-x} dx \\ &=nI_{n-1} \end{aligned} \end{equation}\]
.Repeatedly application of integration by parts gives
\[I_n= \int^{\infty}_0 x^ne^{-x} dx=n! \int^{\infty}_0 e^{-x}= n! [-e^{-x} ]^{\infty}_0 = n! \]
.