Bernoulli's Formula

The indefinite integral $\displaystyle\int u(x)v(x)\,dx$ of a product becomes especially simple to finish in one pass — instead of repeated integration by parts — when $u(x)=a_0x^n+a_1x^{n-1}+\cdots+a_n$ is a polynomial and $v(x)$ can be integrated over and over with ease (e.g. $\sin nx,\cos nx,e^{ax}$). The resulting shortcut is called Bernoulli's formula, an extension of ordinary integration by parts.

Notation. Write the successive derivatives of $u$ as

and the successive anti-derivatives of $v$ as

so that $dv_{(1)}=v\,dx,\ dv_{(2)}=v_{(1)}\,dx,\ dv_{(3)}=v_{(2)}\,dx,\ldots$

Derivation. By integration by parts, $\int uv\,dx=\int u\,dv_{(1)}=uv_{(1)}-\int v_{(1)}\,du=uv_{(1)}-\int u^{(1)}v_{(1)}\,dx=uv_{(1)}-\int u^{(1)}\,dv_{(2)}$. Applying by-parts again to the last integral: $\int u^{(1)}\,dv_{(2)}=u^{(1)}v_{(2)}-\int v_{(2)}\,du^{(1)}=u^{(1)}v_{(2)}-\int u^{(2)}v_{(2)}\,dx$. Substituting back and repeating this process gives

called Bernoulli's formula for the integral of a product.