Integration by Parts on a Closed Interval

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $a,b\in\mathbb{R}$ with $a<b$. Derivatives of maps $[a,b]\to\mathbb{R}$ are taken relative to $[a,b]$, an interval containing two distinct points by Basic Facts about Intervals of the Real Line (v). Let $F,G:[a,b]\to\mathbb{R}$ be differentiable, and suppose that their derivatives $F',G':[a,b]\to\mathbb{R}$ are continuous on $[a,b]$. Let $FG'$ and $F'G$ be the pointwise products, and let integrability on $[a,b]$ and the integral be as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals.

(i) $FG'$ and $F'G$ are bounded and integrable on $[a,b]$.

(ii)

$$\int_{a}^{b}F(t)\,G'(t)\,dt=F(b)\,G(b)-F(a)\,G(a)-\int_{a}^{b}F'(t)\,G(t)\,dt .$$