If $u,v$ are differentiable on $[a,b]$ with $u^{\mathrm{\prime }},v^{\mathrm{\prime }}$ integrable, then ${\int }_a^{b}u{v}^{\mathrm{\prime }}=u(b)v(b)-u(a)v(a)-{\int }_a^b{u}^{\mathrm{\prime }}v$

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 $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ 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 $u^{\mathrm{\prime }}$ and $v^{\mathrm{\prime }}$ 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 $u{v}^{\mathrm{\prime }}$ and $u^{\mathrm{\prime }}v$ are integrable and

${\int }_a^{b}u{v}^{\mathrm{\prime }}=u(b)v(b)-u(a)v(a)-{\int }_a^b{u}^{\mathrm{\prime }}v\mathrm{.}$

The integrability of $u^{\mathrm{\prime }}$ and $v^{\mathrm{\prime }}$ 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).

The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set

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 ${\int }_a^{b}f$

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 $(uv{)}^{\mathrm{\prime }}$ to be integrable. Integrability of $u^{\mathrm{\prime }}$ and $v^{\mathrm{\prime }}$ plus continuity of $u$ and $v$ delivers it, through the product clause of If $f,g$ are integrable on $[a,b]$ then so are $\mid f\mid$, $f^2$, $fg$, $\max (f,g)$ and $\min (f,g)$, and $\mid {\int }_a^{b}f\mid \le {\int }_a^b\mid f\mid$; nothing weaker is used, and nothing weaker is claimed to suffice.

• The boundary term is exactly the increment of $uv$. Writing the identity as ${\int }_a^{b}u{v}^{\mathrm{\prime }}+{\int }_a^b{u}^{\mathrm{\prime }}v=(uv)(b)-(uv)(a)$ 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 ${\int }_a^{b}u{v}^{\mathrm{\prime }}=u(b)v(b)-u(a)v(a)-{\int }_a^b{u}^{\mathrm{\prime }}v$ on the companion page; nothing above depends on it.