Bernoulli's Formula for Integration by Parts

When a product integrand needs integration by parts applied two, three, or more times in a row — most often when one factor is $x^n$ for $n\ge2$ — repeating the ordinary by-parts formula step by step gets long and error-prone. Bernoulli's formula packages the whole repeated process into a single alternating-sign expression.

If $u$ and $v$ are functions of $x$, Bernoulli's rule states:

where $u',u'',u''',\ldots$ are the successive derivatives of $u$ (differentiate $u$ again and again until it becomes $0$, which happens after finitely many steps whenever $u$ is a polynomial), and $v,v_1,v_2,v_3,\ldots$ are the successive integrals of $dv$ (integrate $dv$ once to get $v$, integrate $v\,dx$ again to get $v_1$, and so on). The pattern of signs strictly alternates $+,-,+,-,\ldots$, and the process terminates automatically once a derivative of $u$ reaches $0$ (for $u=x^n$ with $n$ a positive integer, this takes exactly $n+1$ terms).

Bernoulli's formula is most advantageous exactly when $u=x^n$ for a positive integer $n$, since only then does the chain of derivatives $u',u'',\ldots$ terminate in finitely many steps; it saves writing out $n$ separate applications of the ordinary by-parts formula.

Worked-style illustration (paralleling Example 11.35). For $\int x^2e^{5x}\,dx$: take $u=x^2$ (so $u'=2x$, $u''=2$, $u'''=0$) and $dv=e^{5x}dx$ (so $v=\dfrac{e^{5x}}5$, $v_1=\dfrac{e^{5x}}{25}$, $v_2=\dfrac{e^{5x}}{125}$). Bernoulli's formula gives directly

in one line instead of two separate by-parts steps. The same table-based bookkeeping handles $\int x^3\cos x\,dx$ (four terms, since $u=x^3$ needs three differentiations to reach $0$) and $\int x^3e^{-x}\,dx$ equally directly. …