Integration by parts/Fact

Integration by parts

Let

$f,g\colon [a,b]\longrightarrow \mathbb {R}$

denote continuously differentiable functions.

Then

${}\int _{a}^{b}f(t)g'(t)\,dt=fg|_{a}^{b}-\int _{a}^{b}f'(t)g(t)\,dt\,.$

Proof, Write another proof