Proof of Integration by Parts on a Closed Interval

Continuity and integrability. Let $x\in[a,b]$. As $F$ and $G$ are differentiable at $x$, they are continuous at $x$ by The Increment Form of the Derivative, and Differentiability Implies Continuity (ii); as $x$ was arbitrary, $F$ and $G$ are continuous on $[a,b]$ by The Real Line: Standing Notation and Background for Calculus on Intervals (vi). Let $h=F'G+FG':[a,b]\to\mathbb{R}$, the pointwise sum. By Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v), applied with $(X,d)=(\mathbb{R},d_{\mathbb{R}})$ and $A=[a,b]$, the maps $FG'$, $F'G$, $h$ and $FG$ are continuous on $[a,b]$. Each of them is therefore bounded, by Extreme Value Theorem on a Closed Real Interval (i), and integrable on $[a,b]$, by A Continuous Function on a Closed Interval is Riemann Integrable (i). For $FG'$ and $F'G$ this is (i).

An antiderivative of $h$. Let $x\in[a,b]$. By Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (vi), $FG$ is differentiable at $x$ and

$$(FG)'(x)=F'(x)\,G(x)+F(x)\,G'(x)=h(x),$$

the derivative being unique by Derivative of a Real Function on an Interval (ii). In particular $FG$ is continuous on $[a,b]$ and differentiable at every $x\in(a,b)$ with $(FG)'(x)=h(x)$, and $h$ is bounded and integrable on $[a,b]$. By Fundamental Theorem of Calculus, Part II: Evaluating an Integral by an Antiderivative (i), applied to $h$ in place of $f$ and $FG$ in place of $F$,

$$\int_{a}^{b}h=(FG)(b)-(FG)(a)=F(b)\,G(b)-F(a)\,G(a).$$

Conclusion. By Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (i), applied to the bounded integrable maps $F'G$ and $FG'$,

$$\int_{a}^{b}h=\int_{a}^{b}F'G+\int_{a}^{b}FG' .$$

Comparing the two expressions for $\int_{a}^{b}h$ and subtracting $\int_{a}^{b}F'G$ gives

$$\int_{a}^{b}FG'=F(b)\,G(b)-F(a)\,G(a)-\int_{a}^{b}F'G,$$

which, as $(FG')(t)=F(t)\,G'(t)$ and $(F'G)(t)=F'(t)\,G(t)$ for $t\in[a,b]$ by Pointwise Operations on Real-Valued Maps (i), is (ii).