## Reduction Formulae

We can derive iterative formulae for certain integrals to express am integration in terms of a simpler integration. By doing this repeatedly an integral can often be reduced to a very simple integral.
Example:
$I= \int^{\infty}_0 x^ne^{-x} dx$
.
Let
$I_n= \int^{\infty}_0 x^ne^{-x} dx$
. Integration by parts gives
\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}
.
Repeatedly application of integration by parts gives
$I_n= \int^{\infty}_0 x^ne^{-x} dx=n! \int^{\infty}_0 e^{-x}= n! [-e^{-x} ]^{\infty}_0 = n!$
.
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