Something Interesting: A Generalised Rule for Repeated Integration by Parts

Ordinary integration by parts is $\int uv\,dx=u\int v\,dx-\int\left(\dfrac{du}{dx}\int v\,dx\right)dx$, where $u,v$ follow the LIATE order. When $u$ is a polynomial, applying this rule over and over eventually terminates, because some high-enough derivative of a polynomial is exactly zero. Writing $v_1=\int v\,dx$, $v_2=\int v_1\,dx$, $v_3=\int v_2\,dx$, … (repeated integration of $v$) and $u',u'',u''',\dots$ (repeated differentiation of $u$), the whole chain of by-parts steps compresses into one generalised formula:

with the signs alternating, and the process stopping as soon as a derivative of $u$ becomes zero. This is especially convenient whenever $u$ is a polynomial, since only finitely many terms then appear. …