Integration by Parts/Corollary

Corollary to Integration by Parts

Let $u$ and $v$ be real functions which are integrable on their domain.

Then:

$\ds \int u v \rd x = \paren {\int u \rd x} v - \int \paren {\int u \rd x} \dfrac {\d v} {\d x} \rd x$

$(1): \quad \ds \int u \dfrac {\d v} {\d x} \rd x = u v - \int v \dfrac {\d u} {\d x} \rd x$

In $(1)$, let us make the identifications:

$\text {(2)}: \quad$

$\ds u$

$\gets$

$\ds v$

and:

$\ds \dfrac {\d v} {\d x}$

$\gets$

$\ds u$

$\text {(3)}: \quad$

$\ds \leadsto \ \ $

$\ds v = \int \dfrac {\d v} {\d x} \rd x$

$\gets$

$\ds \int u \rd x$

This gives:

$\ds \int u v \rd x$

$=$

$\ds \int v \cdot u \rd x$

changing order for clarity

$\ds $

$=$

$\ds v \int u \rd x - \int \paren {\int u \rd x} \dfrac {\d v} {\d x} \rd x$

substituting from $(2)$ and $(3)$

$\blacksquare$

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Integration by Parts: $3.3.13$