Answer
The product rule
Work Step by Step
\[\begin{align}
& \text{The integration by parts is based on the product rule of derivatives}\text{.} \\
& \text{Demostration:} \\
& \text{Let the functions }u\left( x \right)\text{ and }v\left( x \right)\text{ then,} \\
& \frac{d}{dx}\left[ u\left( x \right)v\left( x \right) \right]=u\left( x \right)v'\left( x \right)+v\left( x \right)u'\left( x \right) \\
& d\left[ u\left( x \right)v\left( x \right) \right]=\left[ u\left( x \right)v'\left( x \right)+v\left( x \right)u'\left( x \right) \right]dx \\
& \text{By integrating both sides},\text{ we can write this rule in terms of an } \\
& \text{indefinite integral}: \\
& u\left( x \right)v\left( x \right)=\int{\left[ u\left( x \right)v'\left( x \right)+v\left( x \right)u'\left( x \right) \right]}dx \\
& \text{Rearranging this expression in the form} \\
& \int{u\left( x \right)}\underbrace{v'\left( x \right)dx}_{dv}=u\left( x \right)v\left( x \right)-\int{v\left( x \right)\underbrace{u'\left( x \right)dx}_{du}} \\
& \int{u}dv=uv-\int{vdu} \\
& \text{} \\
\end{align}\]
