If $u,v$ are differentiable on $[a,b]$ with $u',v'$ integrable, then $\int_a^b u v' = u(b)v(b)-u(a)v(a) - \int_a^b u'v$

Let $a < b$ be reals and let $u, v : [a,b] \to \mathbb{R}$ be differentiable at every point of $[a,b]$ as functions on $[a,b]$ (The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set). Suppose $u'$ and $v'$ are integrable on $[a,b]$ (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$). Then $uv'$ and $u'v$ are integrable and

$\int_a^b u\,v' \;=\; u(b)v(b) - u(a)v(a) \;-\; \int_a^b u'\,v .$

The integrability of $u'$ and $v'$ is a hypothesis, not a formality. Without it the two integrals in the display need not exist at all, and the identity is then not false but ill-formed; that is the false statement that deletes it on the companion page. The hypothesis is automatic when $u$ and $v$ are continuously differentiable, since a continuous function on $[a,b]$ is integrable (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

• Step 2.1 is the step usually skipped, and it is why the hypotheses are what they are. The identity is an application of the second fundamental theorem to $uv$, and that theorem needs $(uv)'$ to be integrable. Integrability of $u'$ and $v'$ plus continuity of $u$ and $v$ delivers it, through the product clause of If $f,g$ are integrable on $[a,b]$ then so are $\lvert f\rvert$, $f^{2}$, $fg$, $\max(f,g)$ and $\min(f,g)$, and $\bigl\lvert\int_a^b f\bigr\rvert \le \int_a^b\lvert f\rvert$; nothing weaker is used, and nothing weaker is claimed to suffice.

• The boundary term is exactly the increment of $uv$. Writing the identity as $\int_a^b uv' + \int_a^b u'v = (uv)(b) - (uv)(a)$ makes the symmetry in $u$ and $v$ visible and is the form worth remembering.

• Discrete counterpart. Abel's summation by parts (Abel summation by parts: with $A_n = \sum_{k<n} a_k$ one has $\sum_{k<n} a_k b_k = A_n b_{n-1} - \sum_{k < n-1} A_{k+1}\,(b_{k+1} - b_k)$ for every $n \ge 1$) is the same manipulation for finite sums, and it is what Bonnet's second mean value theorem: for $f$ monotone and $g$ integrable on $[a,b]$ there is $\xi\in[a,b]$ with $\int_a^b fg = f(a)\int_a^\xi g + f(b)\int_\xi^b g$ below uses in place of this theorem, precisely because that theorem assumes no differentiability.

• Forward reference, orientation only. The false statement that deletes the integrability hypothesis is FALSE: if $u$ and $v$ are differentiable on $[a,b]$ then $\int_a^b uv' = u(b)v(b)-u(a)v(a)-\int_a^b u'v$ ↗ on the companion page; nothing above depends on it.