Question

how the integration by parts formula related to the product rule of differentiation?

Answer 1

Product rule:
\[(uv)' = uv' + vu'\]
in other words
\[\frac{d}{dx} (uv) = udv + vdu\]
subtract vdu from both sides
\[\implies \frac{d}{dx} (uv) - vdu = udv\]
integrate both sides
\[\implies \int \frac{d}{dx} (uv) - \int vdu = \int udv\]
simplify
\[\implies uv - \int vdu = \int udv\]
look familiar?