The tutor shows the motivation behind the integration by parts formula.

Integration by parts is a reversal of the product rule (see my post here).

If we start with the product rule as

(uv)’=uv’ + u’v

then we integrate both sides, we get

∫(uv)’=∫uv’ + ∫u’v

Integral is opposite of derivative, so ∫(uv)’ = uv:

uv = ∫uv’ + ∫u’v

More conveniently, it can be written

∫uv’ + ∫u’v = uv

Next, subtracting ∫u’v from both sides, we have

∫uv’ = uv – ∫u’v

The last term is usually written ∫vu’:

∫uv’ = uv – ∫vu’

so that we have the familiar formula for integration by parts.

In a coming post I’ll show an example of how to use the integration by parts formula.

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus, third edition. Toronto:
  D C Heath and Company, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.