Integration by parts/Fact/Proof

Proof

Due to the product rule, the function ${}fg$ is a primitive function for ${}fg'+f'g$. Therefore,

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

To fact