How would you prove the 'integration by parts' rule?

This involves thinking about a well-known formula (the product rule) in a slightly different way. Looking at the product rule, for two functions u and v, (uv)' = uv' + vu'. We can rewrite this as uv' = (uv)' - vu'. Integrating both sides, we obtain integral of uv' = uv - integral of vu'.